Topological space: Difference between revisions

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imported>Hendra I. Nurdin
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imported>Jitse Niesen
(merge text from Topological Space, add how a metric defines a topology, add alternative definition through neighbourhoods)
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==Formal definition==  
==Formal 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. <math> A \in O \Rightarrow A \subset X</math>) with the following three properties:
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:


1. <math>X</math> and <math>\emptyset</math> (the empty set) are in <math>O</math>
# <math>X</math> and <math>\emptyset</math> (the empty set) are in <math>O</math>
 
# The union of any number (countable or uncountable) of elements of <math>O</math> is again in <math>O</math>     
2. The union of any number (countable or uncountable) of elements of <math>O</math> is again in <math>O</math>     
# The intersection of any ''finite'' number of elements of <math>O</math> is again in <math>O</math>   
 
3. The intersection of any <i>finite</i> number of elements of <math>O</math> is again in <math>O</math>   


Elements of the set <math>O</math> are called open sets (of <math>X</math>).  
Elements of the set <math>O</math> are called open sets (of <math>X</math>).  
   
   
Note that as shorthand a topological space <math>(X,O)</math> is often simply written as <math>X</math> once the particular topology on <math>X</math> is understood.
A topological space <math>(X,O)</math> is often simply written as <math>X</math> once the particular topology on <math>X</math> is understood.


== Examples ==
== Examples ==
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1. Let <math>X=\mathbb{R}</math> where <math>\mathbb{R}</math> denotes the set of real numbers. The open interval ]''a'', ''b''[ (where ''a'' < ''b'') is the set of all numbers between ''a'' and ''b'':
1. Let <math>X=\mathbb{R}</math> where <math>\mathbb{R}</math> denotes the set of real numbers. The open interval ]''a'', ''b''[ (where ''a'' < ''b'') is the set of all numbers between ''a'' and ''b'':


<center><math> \mathopen{]} a,b \mathclose{[} = \{ y \in \mathbb{R} \mid a < y < b \}.</math></center>  
:<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>O</math> can be defined on <math>X=\mathbb{R}</math> to consist of <math>\emptyset</math> and all sets of the form:  


<center><math>\bigcup_{\gamma \in \Gamma} \mathopen{]} a_\gamma, b_\gamma \mathclose{[} ,</math></center>  
:<math>\bigcup_{\gamma \in \Gamma} \mathopen{]} a_\gamma, b_\gamma \mathclose{[} ,</math>
 
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''.
 
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
:<math> B_r(x) = \{ y \in X \mid d(x,y) < r \}. </math>
A set <math>A \subset X</math> is open if and only if for every <math>x \in A</math>, there is an open ball with centre <math>x</math> contained in <math>A</math>. The resulting topology is called the topology induced by the metric <math>d</math>. The standard topology on <math>\mathbb{R}</math>, discussed in Example 1, is induced by the metric <math>d(x,y) = |x-y|</math>.
 
== Neighbourhoods ==
Given a topological space <math>(X,O)</math>, we say that a subset ''N'' of ''X'' is a neighbourhood of a point <math>x \in X</math> if ''N'' contains an open set <math>U \in O</math> containing the point ''x''.<ref>Some authors use a different definition, in which a neighbourhood ''N'' of ''x'' is an open set containing ''x''.</ref>


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 topology is precisely the familiar topology induced on <math>\mathbb{R}</math> by the Euclidean distance <math>d(x,y)=|x-y|</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.  
In this article, we defined topological spaces in terms of their open sets. It is also possible to define topological spaces in terms of neighbourhoods. In this alternative approach, a topological space is defined by the set <math>X</math> and for each <math>x \in X</math>, a collection <math>N_x</math> of subsets of <math>X</math> (the neighbourhoods of <math>x</math>), such that:
# <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>
# The intersection of two elements of <math>N_x</math> is again in <math>N_x</math>
# If <math>U</math> is in <math>N_x</math> and <math>V \subset X</math> contains <math>U</math>, then <math>V</math> is again in <math>N_x</math>
# If <math>U</math> is in <math>N_x</math> then there exists a <math>V \in N_x</math> such that <math>V</math> is a subset of <math>U</math> and <math>V \in N_y</math> for all <math>y \in V</math>
The open sets in this topology are precisely those sets <math>A</math> that are in <math>N_x</math> for all <math>x \in A</math>.


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 <i>Zariski topology</i>.
This alternative approach is particularly useful when one considers topologies on topological abelian groups and topological rings by subgroups or ideals, respectively, because knowing the neighbourhoods of any point is equivalent to knowing the neighbourhoods of 0.


== Some topological notions==
== Some topological notions==
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; Partial list of topological notions  
; Partial list of topological notions  
; Neighbourhood : A subset ''N'' of ''X'' is a neighbourhood of a point <math>x \in X</math> if ''N'' contains an open set <math>U \in O</math> containing the point ''x'' 
; 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 ''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''.
; 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 ''X'' if each point <math>x \in X</math> belongs to at least one of the open sets in <math>\mathcal{U}</math>  
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== See also ==
== See also ==


[[Topology]]
* [[Topology]]
 
* [[Analysis]]
[[Analysis]]
* [[Metric space]]


[[Metric space]]
== Notes ==
<references/>


== References ==
== References ==


1. K. Yosida, <i>Functional Analysis</i> (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980
# K. Yosida, ''Functional Analysis'' (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980
 
# D. Wilkins, Lecture notes for Course 212 - Topology, Trinity College Dublin, URL: [http://www.maths.tcd.ie/~dwilkins/Courses/212/]   
2. D. Wilkins, Lecture notes for Course 212 - Topology, Trinity College Dublin, URL: [http://www.maths.tcd.ie/~dwilkins/Courses/212/]   


[[Category:Mathematics Workgroup]]
[[Category:Mathematics Workgroup]]
[[Category:CZ Live]]
[[Category:CZ Live]]

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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 introduces a structure on the set which is useful for defining some important abstract notions such as the "closeness" of two elements of and convergence of sequences of elements of .

Formal definition

A topological space is an ordered pair where is a set and is a collection of subsets of (i.e., any element is a subset of X) with the following three properties:

  1. and (the empty set) are in
  2. The union of any number (countable or uncountable) of elements of is again in
  3. The intersection of any finite number of elements of is again in

Elements of the set are called open sets (of ).

A topological space is often simply written as once the particular topology on is understood.

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:

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 \mathopen{]} a,b \mathclose{[} = \{ y \in \mathbb{R} \mid a < y < b \}.}

Then a 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 O} can be defined on to consist 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 \emptyset} and all 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 \bigcup_{\gamma \in \Gamma} \mathopen{]} a_\gamma, b_\gamma \mathclose{[} ,}

where is any arbitrary index 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 a_{\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 b_{\gamma}} are real numbers satisfying 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_\gamma < b_\gamma} 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 \gamma \in \Gamma } . 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 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 in the next example another 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}} , albeit a relatively obscure one, will be constructed.

2. Let 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=\mathbb{R}} as before. Let 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 O} be a collection of subsets 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 \mathbb{R}} defined by the requirement that 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 A=\emptyset} 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 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|} .

Neighbourhoods

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,O)} , we say that a subset N of X is a neighbourhood 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 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 O} containing the point x.[1]

In this article, we defined topological spaces in terms of their open sets. It is also possible to define topological spaces in terms of neighbourhoods. In this alternative approach, a topological space is defined by 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} and for each 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} , 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 N_x} of subsets 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} (the neighbourhoods 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} ), such that:

  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 V \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}

The open sets in this topology are precisely those 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 A} that are 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 A} .

This alternative approach is particularly useful when one considers topologies on topological abelian groups and topological rings by subgroups or ideals, respectively, because knowing the neighbourhoods of any point is equivalent to knowing the neighbourhoods of 0.

Some topological notions

This section introduces some important topological notions. Throughout, X will denote a topological space with the topology O.

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 A of X if any open set in O containing 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 A if every neighbourhood of 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 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 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
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 U and 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
X is connected if given any two disjoint open sets U and 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 X=U or X=V
Path-connectedness
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 x to y
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

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,O_X)} will denote a topological space.

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 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 O \cap A} where O 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 O_X} . The collection of all such open sets defines a topology on A called the relative topology of A as a subset of X
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 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 O_X} . The collection of all such open sets defines a topology on Y called the quotient topology induced by q

See also

Notes

  1. ↑ Some authors use a different definition, in which a neighbourhood 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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