Topological space: Difference between revisions

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imported>Wlodzimierz Holsztynski
(→‎Definition: "of open sets" was not only clumsy but even wrong (in topology we say "a space of POINTS", not "of open sets").)
imported>Andrey Khalyavin
(Changed O to T. T is more convenient in definition of the topological space.+math tags)
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In [[mathematics]], a '''topological space''' is an ordered pair <math>(X,O)</math> where <math>X</math> is a set and <math>O</math> is a certain collection of subsets of <math>X</math> called the <i>open sets</i> or the <i>topology</i> of <math>X</math>. The topology of <math>X</math> introduces a structure on the set <math>X</math> which is useful for defining some important abstract notions such as the "closeness" of two elements of <math>X</math> and [[convergence of sequences]] of elements of <math>X</math>.
In [[mathematics]], a '''topological space''' is an ordered pair <math>(X,\mathcal T)</math> where <math>X</math> is a set and <math>\mathcal T</math> is a certain collection of subsets of <math>X</math> called the <i>open sets</i> or the <i>topology</i> of <math>X</math>. The topology of <math>X</math> introduces a structure on the set <math>X</math> which is useful for defining some important abstract notions such as the "closeness" of two elements of <math>X</math> and [[convergence of sequences]] of elements of <math>X</math>.


==Definition==  
==Definition==  
A topological space is an ordered pair <math>(X,O)</math> where <math>X</math> is a set and <math>O</math> is a collection of subsets of <math>X</math> (i.e., any element <math> A \in O </math> is a subset of ''X'') with the following three properties:
A topological space is an ordered pair <math>(X,\mathcal T)</math> where <math>X</math> is a set and <math>\mathcal T</math> is a collection of subsets of <math>X</math> (i.e., any element <math> A \in \mathcal T </math> is a subset of ''X'') with the following three properties:


# <math>X</math> and <math>\varnothing</math> (the empty set) are in <math>O</math>
# <math>X</math> and <math>\varnothing</math> (the empty set) are in <math>\mathcal T</math>
# The union of any family (infinite or otherwise) of elements of <math>O</math> is again in <math>O</math>     
# The union of any family (infinite or otherwise) of elements of <math>\mathcal T</math> is again in <math>\mathcal T</math>     
# The intersection of two elements of <math>O</math> is again in <math>O</math>   
# The intersection of two elements of <math>\mathcal T</math> is again in <math>\mathcal T</math>   


Elements of the set <math>O</math> are called open sets of <math>X</math>.  We often simply write <math>X</math> instead of <math>(X,O)</math> once the topology <math>O</math> is established.
Elements of the set <math>\mathcal T</math> are called open sets of <math>X</math>.  We often simply write <math>X</math> instead of <math>(X,\mathcal T)</math> once the topology <math>\mathcal T</math> is established.


Once we have a topology on <math>X</math>, we define the ''closed sets'' of <math>X</math> to be the compliments (in <math>X</math>) of the open sets; the closed sets of <math>X</math> have the following characteristic properties:
Once we have a topology on <math>X</math>, we define the ''closed sets'' of <math>X</math> to be the compliments (in <math>X</math>) of the open sets; the closed sets of <math>X</math> have the following characteristic properties:
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# The union of two closed sets is closed
# The union of two closed sets is closed


Alternatively, notice that we could have defined a structure of closed sets (having the properties above as axioms) and defined the open sets relative to that structure as compliments of closed sets. Then such a family of open sets obeys the axioms for a topology; we obtain a one to one correspondence between topologies and structures of closed sets. Similarly, the axioms for systems of neighborhoods (described below) give rise to a collection of "open sets" verifying the axioms for a topology, and conversely--every topology defines the systems of neighborhoods; for every set ''X'' we obtain a one to one correspondence between topologies in X and systems of neighborhoods in X.  These correspondences allow one to study the topological structure from different viewpoints.
Alternatively, notice that we could have defined a structure of closed sets (having the properties above as axioms) and defined the open sets relative to that structure as compliments of closed sets. Then such a family of open sets obeys the axioms for a topology; we obtain a one to one correspondence between topologies and structures of closed sets. Similarly, the axioms for systems of neighborhoods (described below) give rise to a collection of "open sets" verifying the axioms for a topology, and conversely --- every topology defines the systems of neighborhoods; for every set ''X'' we obtain a one to one correspondence between topologies in X and systems of neighborhoods in X.  These correspondences allow one to study the topological structure from different viewpoints.


==The category of topological spaces==
==The category of topological spaces==
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:<math> \mathopen{]} a,b \mathclose{[} = \{ y \in \mathbb{R} \mid a < y < b \}.</math>
:<math> \mathopen{]} a,b \mathclose{[} = \{ y \in \mathbb{R} \mid a < y < b \}.</math>


Then a topology <math>O</math> can be defined on <math>X=\mathbb{R}</math> to consist of <math>\emptyset</math> and all sets of the form:  
Then a topology <math>\mathcal T</math> can be defined on <math>X=\mathbb{R}</math> to consist of <math>\emptyset</math> and all sets of the form:  


:<math>\bigcup_{\gamma \in \Gamma} \mathopen{]} a_\gamma, b_\gamma \mathclose{[} ,</math>
:<math>\bigcup_{\gamma \in \Gamma} \mathopen{]} a_\gamma, b_\gamma \mathclose{[} ,</math>
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where <math>\Gamma</math> is any arbitrary index set, and <math>a_{\gamma}</math> and <math>b_{\gamma}</math> are real numbers satisfying <math>a_\gamma < b_\gamma</math> for all <math>\gamma \in \Gamma </math>. This is the familiar topology on <math>\mathbb{R}</math> and probably the most widely used in the applied sciences. However, in general one may define different inequivalent topologies on a particular set <math>X</math> and in the next example another topology on <math>\mathbb{R}</math>, albeit a relatively obscure one,  will be constructed.  
where <math>\Gamma</math> is any arbitrary index set, and <math>a_{\gamma}</math> and <math>b_{\gamma}</math> are real numbers satisfying <math>a_\gamma < b_\gamma</math> for all <math>\gamma \in \Gamma </math>. This is the familiar topology on <math>\mathbb{R}</math> and probably the most widely used in the applied sciences. However, in general one may define different inequivalent topologies on a particular set <math>X</math> and in the next example another topology on <math>\mathbb{R}</math>, albeit a relatively obscure one,  will be constructed.  


2. Let <math>X=\mathbb{R}</math> as before. Let <math>O</math> be a collection of subsets of <math>\mathbb{R}</math> defined by the requirement that <math>A \in O </math> if and only if <math>A=\emptyset</math> or <math>A</math> contains all except at most a finite number of real numbers. Then it is straightforward to verify that <math>O</math> defined in this way has the three properties required to be a topology on <math>\mathbb{R}</math>. This topology is known as the ''cofinite topology'' or ''Zariski topology''.
2. Let <math>X=\mathbb{R}</math> as before. Let <math>\mathcal T</math> be a collection of subsets of <math>\mathbb{R}</math> defined by the requirement that <math>A \in \mathcal T </math> if and only if <math>A=\emptyset</math> or <math>A</math> contains all except at most a finite number of real numbers. Then it is straightforward to verify that <math>\mathcal T</math> defined in this way has the three properties required to be a topology on <math>\mathbb{R}</math>. This topology is known as the ''cofinite topology'' or ''Zariski topology''.


3. Every [[metric space|metric]] <math>d</math> on <math>X</math> gives rise to a topology on <math>X</math>. The open ball with centre <math>x \in X</math> and radius <math>r > 0</math> is defined to be the set
3. Every [[metric space|metric]] <math>d</math> on <math>X</math> gives rise to a topology on <math>X</math>. The open ball with centre <math>x \in X</math> and radius <math>r > 0</math> is defined to be the set
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== Neighborhoods ==
== Neighborhoods ==
Given a topological space <math>(X,O)</math> of opens, we say that a subset <math>N</math> of <math>X</math> is a ''neighborhood'' of a point <math>x \in X</math> if <math>N</math> contains an open set <math>U \in O</math> containing the point <math>x</math> <ref>Some authors use a different definition, in which a neighborhood ''N'' of ''x'' is an open set containing ''x''.</ref>  
Given a topological space <math>(X,\mathcal T)</math> of opens, we say that a subset <math>N</math> of <math>X</math> is a ''neighborhood'' of a point <math>x \in X</math> if <math>N</math> contains an open set <math>U \in \mathcal T</math> containing the point <math>x</math> <ref>Some authors use a different definition, in which a neighborhood ''N'' of ''x'' is an open set containing ''x''.</ref>  


If <math>N_x</math> denotes the system of neighborhoods of <math>x</math> relative to the topology <math>O</math>, then the following properties hold:
If <math>N_x</math> denotes the system of neighborhoods of <math>x</math> relative to the topology <math>\mathcal T</math>, then the following properties hold:
# <math>N_x</math> is not empty for any <math>x \in X</math>
# <math>N_x</math> is not empty for any <math>x \in X</math>
# If <math>U</math> is in <math>N_x</math> then <math>x \in U</math>
# If <math>U</math> is in <math>N_x</math> then <math>x \in U</math>
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== Some topological notions==
== Some topological notions==
This section introduces some important topological notions. Throughout, ''X'' will denote a topological space with the topology ''O''.  
This section introduces some important topological notions. Throughout, <math>X</math> will denote a topological space with the topology <math>\mathcal T</math>.  


; Partial list of topological notions  
; Partial list of topological notions  
; Limit point : A point <math>x \in X</math> is a limit point of a subset ''A'' of ''X'' if any open set in ''O'' containing ''x'' also contains a point <math>y \in A</math> with <math>y \ne x</math>. An equivalent definition is that <math>x \in X</math> is a limit point of ''A'' if every neighbourhood of ''x'' contains a point <math>y \in A</math> different from ''x''.
; Limit point : A point <math>x \in X</math> is a limit point of a subset <math>A</math> of <math>X</math> if any open set in <math>\mathcal T</math> containing <math>x</math> also contains a point <math>y \in A</math> with <math>y \ne x</math>. An equivalent definition is that <math>x \in X</math> is a limit point of <math>A</math> if every neighbourhood of <math>x</math> contains a point <math>y \in A</math> different from <math>x</math>.
; Open cover : A collection <math>\mathcal{U}</math> of open sets of ''X'' is said to be an open cover for ''X'' if each point <math>x \in X</math> belongs to at least one of the open sets in <math>\mathcal{U}</math>  
; Open cover : A collection <math>\mathcal{U}</math> of open sets of ''X'' is said to be an open cover for <math>X</math> if each point <math>x \in X</math> belongs to at least one of the open sets in <math>\mathcal{U}</math>  
; Path: A path <math>\gamma</math> is a [[continuous function]] <math>\gamma:[0,1]\rightarrow X</math>. The point <math>\gamma(0)</math> is said to be the '''starting point''' of <math>\gamma</math> and <math>\gamma(1)</math> is said to be the '''end point'''. A path joins its starting point to its end point
; Path: A path <math>\gamma</math> is a [[continuous function]] <math>\gamma:[0,1]\rightarrow X</math>. The point <math>\gamma(0)</math> is said to be the '''starting point''' of <math>\gamma</math> and <math>\gamma(1)</math> is said to be the '''end point'''. A path joins its starting point to its end point
; Hausdorff/separability property: ''X'' has the Hausdorff (or separability) property if for any pair <math>x,y \in X</math> there exist ''disjoint'' sets ''U'' and ''V'' with <math>x \in U</math> and <math>y \in V</math>  
; Hausdorff/separability property: <math>X</math> has the Hausdorff (or separability) property if for any pair <math>x,y \in X</math> there exist ''disjoint'' sets <math>U</math> and <math>V</math> with <math>x \in U</math> and <math>y \in V</math>  
; Connectedness: ''X'' is connected if given any two ''disjoint'' open sets ''U'' and ''V'' such that <math>X=U \cup V </math>, then either ''X=U'' or ''X=V''
; Connectedness: <math>X</math> is connected if given any two ''disjoint'' open sets <math>U</math> and <math>V</math> such that <math>X=U \cup V </math>, then either <math>X=U</math> or <math>X=V</math>
; Path-connectedness : ''X'' is path-connected if for any pair <math>x,y \in X</math> there exists a path joining ''x'' to ''y''
; Path-connectedness : <math>X</math> is path-connected if for any pair <math>x,y \in X</math> there exists a path joining <math>x</math> to <math>y</math>
; Compactness : ''X'' is said to be compact if any open cover of ''X'' has a ''finite sub-cover''. That is, any open cover has a finite number of elements which again constitute an open cover for ''X''    
; Compactness : <math>X</math> is said to be compact if any open cover of <math>X</math> has a ''finite sub-cover''. That is, any open cover has a finite number of elements which again constitute an open cover for <math>X</math>    


A topological space with the Hausdorff, connectedness, path-connectedness property is called, respectively, a Hausdorff (or separable), connected, path-connected topological space. A path connected topological space is also connected, but the converse need not be true.
A topological space with the Hausdorff, connectedness, path-connectedness property is called, respectively, a Hausdorff (or separable), connected, path-connected topological space. A path connected topological space is also connected, but the converse need not be true.


==Induced topologies==
==Induced topologies==
A topological space can be used to define a topology on any particular subset or on another set. These "derived" topologies are referred to as '''induced topologies'''. Descriptions of some induced topologies are given below. Throughout, <math>(X,O_X)</math> will denote a topological space.
A topological space can be used to define a topology on any particular subset or on another set. These "derived" topologies are referred to as '''induced topologies'''. Descriptions of some induced topologies are given below. Throughout, <math>(X,\mathcal T_X)</math> will denote a topological space.


; Some induced topologies  
; Some induced topologies  
; Relative topology : If ''A'' is a subset of ''X'' then open sets may be defined on ''A'' as sets of the form <math>O \cap A</math> where ''O'' is any open set in <math>O_X</math>. The collection of all such open sets defines a topology on ''A'' called the ''relative topology'' of ''A'' as a subset of ''X''    
; Relative topology : If <math>A</math> is a subset of <math>X</math> then open sets may be defined on <math>A</math> as sets of the form <math>U \cap A</math> where <math>U</math> is any open set in <math>\mathcal T_X</math>. The collection of all such open sets defines a topology on <math>A</math> called the ''relative topology'' of <math>A</math> as a subset of <math>X</math>    
; Quotient topology : If ''Y'' is another set and ''q'' is a surjective function from ''X'' to ''Y'' then open sets may be defined on ''Y'' as subsets ''U'' of ''Y'' such that <math>q^{-1}(U)=\{x \in X \mid q(x) \in U \} \in O_X</math>. The collection of all such open sets defines a topology on ''Y'' called the ''quotient topology'' induced by ''q''
; Quotient topology : If <math>Y</math> is another set and <math>q</math> is a surjective function from <math>X</math> to <math>Y</math> then open sets may be defined on <math>Y</math> as subsets <math>U</math> of <math>Y</math> such that <math>q^{-1}(U)=\{x \in X \mid q(x) \in U \} \in \mathcal T_X</math>. The collection of all such open sets defines a topology on <math>Y</math> called the ''quotient topology'' induced by <math>q</math>


== See also ==
== See also ==

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In mathematics, a topological space is an ordered pair where is a set and is a certain collection of subsets of called the open sets or the topology of . The topology of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} introduces a structure on the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} which is useful for defining some important abstract notions such as the "closeness" of two elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and convergence of sequences of elements of .

Definition

A topological space is an ordered pair Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X,\mathcal T)} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is a set and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T} is a collection of subsets of (i.e., any element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \in \mathcal T } is a subset of X) with the following three properties:

  1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varnothing} (the empty set) are in
  2. The union of any family (infinite or otherwise) of elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T} is again in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T}
  3. The intersection of two elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T} is again in

Elements of the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T} are called open sets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} . We often simply write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} instead of once the topology Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T} is established.

Once we have a topology on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} , we define the closed sets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} to be the compliments (in ) of the open sets; the closed sets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} have the following characteristic properties:

  1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varnothing} (the empty set) are closed
  2. The intersection of any family of closed sets is closed
  3. The union of two closed sets is closed

Alternatively, notice that we could have defined a structure of closed sets (having the properties above as axioms) and defined the open sets relative to that structure as compliments of closed sets. Then such a family of open sets obeys the axioms for a topology; we obtain a one to one correspondence between topologies and structures of closed sets. Similarly, the axioms for systems of neighborhoods (described below) give rise to a collection of "open sets" verifying the axioms for a topology, and conversely --- every topology defines the systems of neighborhoods; for every set X we obtain a one to one correspondence between topologies in X and systems of neighborhoods in X. These correspondences allow one to study the topological structure from different viewpoints.

The category of topological spaces

Given that topological spaces capture notions of geometry, a good notion of isomorphism in the category of topological spaces should require that equivalent spaces have equivalent topologies. The correct definition of morphisms in the category of topological spaces is the continuous homomorphism.

A function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:X\to Y} is continuous if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(V)} is open in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} for every open in . Continuity can be shown to be invariant with respect to the representation of the underlying topology; e.g., if is closed in for each closed subset of Y, then is continuous in the sense just defined, and conversely.

Isomorphisms in the category of topological spaces (often referred to as a homeomorphism) are bijective and continuous with continuous inverses.

The category of topological spaces has a number of nice properties; there is an initial object (the empty set), subobjects (the subspace topology) and quotient objects (the quotient topology), and products and coproducts exist as well. The necessary topologies to define on the latter two objects become clear immediately; if they're going to be universal in the category of topological spaces, then the topologies should be the coarsest making the canonical maps commute.

Examples

1. Let where denotes the set of real numbers. The open interval ]a, b[ (where a < b) is the set of all numbers between a and b:

Then a topology can be defined on to consist of and all sets of the form:

where is any arbitrary index set, and and are real numbers satisfying for all . This is the familiar topology on and probably the most widely used in the applied sciences. However, in general one may define different inequivalent topologies on a particular set and in the next example another topology on , albeit a relatively obscure one, will be constructed.

2. Let as before. Let be a collection of subsets of defined by the requirement that if and only if or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} contains all except at most a finite number of real numbers. Then it is straightforward to verify that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T} defined in this way has the three properties required to be a topology on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} . This topology is known as the cofinite topology or Zariski topology.

3. Every metric Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} gives rise to a topology on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} . The open ball with centre Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X} and radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r > 0} is defined to be the set

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_r(x) = \{ y \in X \mid d(x,y) < r \}. }

A set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \subset X} is open if and only if for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in A} , there is an open ball with centre Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} contained in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} . The resulting topology is called the topology induced by the metric Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} . The standard topology on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} , discussed in Example 1, is induced by the metric Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,y) = |x-y|} .

Neighborhoods

Given a topological space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X,\mathcal T)} of opens, we say that a subset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is a neighborhood of a point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} contains an open set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U \in \mathcal T} containing the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} [1]

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_x} denotes the system of neighborhoods of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} relative to the topology Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T} , then the following properties hold:

  1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_x} is not empty for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X}
  2. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} is in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_x} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in U}
  3. The intersection of two elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_x} is again in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_x}
  4. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} is in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V \subset X} contains Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is again in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_x}
  5. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} is in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_x} then there exists a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V \in N_x} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is a subset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U \in N_y} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y \in V}

Conversely, if we define a topology of neighborhoods on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} via the above properties, then we can recover a topology of opens whose neighborhoods relative to that topology give rise to the neighborhood topology we started from: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} is open if it is in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_x} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in U} . Moreover, the opens relative to a topology of neighborhoods form a topology of opens whose neighborhoods are the same as those we started from. All this just means that a given topological space is the same, regardless of which axioms we choose to start from.

The neighborhood axioms lend themselves especially well to the study of topological abelian groups and topological rings because knowing the neighborhoods of any point is equivalent to knowing the neighborhoods of 0 (since the operations are presumed continuous). For example, the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} -adic topology on a ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is Hausdorff if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bigcap I^n=0} , thus a topological property is equivalent to an algebraic property which becomes clear when thinking in terms of neighborhoods.

Some topological notions

This section introduces some important topological notions. Throughout, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} will denote a topological space with the topology Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T} .

Partial list of topological notions
Limit point
A point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X} is a limit point of a subset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} if any open set in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T} containing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} also contains a point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y \in A} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y \ne x} . An equivalent definition is that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X} is a limit point of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} if every neighbourhood of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} contains a point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y \in A} different from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} .
Open cover
A collection Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{U}} of open sets of X is said to be an open cover for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} if each point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X} belongs to at least one of the open sets in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{U}}
Path
A path Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} is a continuous function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma:[0,1]\rightarrow X} . The point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma(0)} is said to be the starting point of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma(1)} is said to be the end point. A path joins its starting point to its end point
Hausdorff/separability property
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} has the Hausdorff (or separability) property if for any pair Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,y \in X} there exist disjoint sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in U} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y \in V}
Connectedness
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is connected if given any two disjoint open sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X=U \cup V } , then either Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X=U} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X=V}
Path-connectedness
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is path-connected if for any pair Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,y \in X} there exists a path joining to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y}
Compactness
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is said to be compact if any open cover of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} has a finite sub-cover. That is, any open cover has a finite number of elements which again constitute an open cover for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X}

A topological space with the Hausdorff, connectedness, path-connectedness property is called, respectively, a Hausdorff (or separable), connected, path-connected topological space. A path connected topological space is also connected, but the converse need not be true.

Induced topologies

A topological space can be used to define a topology on any particular subset or on another set. These "derived" topologies are referred to as induced topologies. Descriptions of some induced topologies are given below. Throughout, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X,\mathcal T_X)} will denote a topological space.

Some induced topologies
Relative topology
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is a subset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} then open sets may be defined on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} as sets of the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U \cap A} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} is any open set in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T_X} . The collection of all such open sets defines a topology on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} called the relative topology of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} as a subset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X}
Quotient topology
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} is another set and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q} is a surjective function from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} then open sets may be defined on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} as subsets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-1}(U)=\{x \in X \mid q(x) \in U \} \in \mathcal T_X} . The collection of all such open sets defines a topology on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} called the quotient topology induced by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q}

See also

Notes

  1. ↑ Some authors use a different definition, in which a neighborhood N of x is an open set containing x.

References

  1. K. Yosida, Functional Analysis (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980
  2. D. Wilkins, Lecture notes for Course 212 - Topology, Trinity College Dublin, URL: [1]
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