Difference between revisions of "Topological space"

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A topological space is a set <math>X</math> coupled with a topology on <math>X</math> denoted <math>\mathcal{J}\subset\mathcal{P}(X)</math>, which is a collection of subsets of <math>X</math> with the following properties:
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{{Stub page|grade=A|msg=Hasn't been updated since March 2015, in April 2016 it was updated to modern format and cleaned up}}
 
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==[[Topological space/Definition|Definition]]==
# Both <math>\emptyset,X\in\mathcal{J}</math>
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{{:Topological space/Definition}}
# For the collection <math>\{U_\alpha\}_{\alpha\in I}\subset\mathcal{J}</math> where <math>I</math> is any indexing set, <math>\cup_{\alpha\in I}U_\alpha\in\mathcal{J}</math> - that is it is closed under union (infinite, finite, whatever)
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==Comparing topologies==
# For the collection <math>\{U_i\}^n_{i=1}\subset\mathcal{J}</math> (any finite collection of members of the topology) that <math>\cap^n_{i=1}U_i\in\mathcal{J}</math>
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Given two ''topological spaces'', {{M|(X_1,\mathcal{J}_1)}} and {{M|(X_2,\mathcal{J}_2)}} we may be able to compare them; we say:
 
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{| class="wikitable" border="1"
We write the topological space as <math>(X,\mathcal{J})</math> or just <math>X</math> if the topology on <math>X</math> is obvious.
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|-
 
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! Terminology
The elements of <math>\mathcal{J}</math> are defined to be "[[Open set|open]]" sets.
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! If
 
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! Comment
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|-
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! {{Anchor|Coarser|Smaller|Weaker}}{{M|\mathcal{J}_1}} coarser{{rITTMJML}}/smaller/weaker {{M|\mathcal{J}_2}}
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| {{M|\mathcal{J}_1\subseteq\mathcal{J}_2}}
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| Using the [[implies-subset relation]] we see that {{M|\mathcal{J}_1\subseteq\mathcal{J}_2\iff\forall S\in\mathcal{J}_1[S\in\mathcal{J}_2]}}
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|-
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! {{Anchor|Finer|Larger|Stronger}}{{M|\mathcal{J}_1}} finer{{rITTMJML}}/larger/stronger {{M|\mathcal{J}_2}}
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| {{M|\mathcal{J}_2\subseteq\mathcal{J}_1}}
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| Again, same idea, {{M|\mathcal{J}_2\subseteq\mathcal{J}_2\iff\forall S\in\mathcal{J}_2[S\in\mathcal{J}_1]}}
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|}
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{{Requires references|grade=C|Need references for larger/smaller/stronger/weaker, Check Introduction To Topology - Mendelson, Addendum: investigate relating this to a [[poset]] (easy enough - not very useful / lacking practical applications)}}
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==Examples==
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* Every [[Metric space|metric space]] induces a topology, see [[Topology induced by a metric|the topology induced by a metric space]]
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* Given any set {{M|X}} we can always define the following two topologies on it:
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*# [[Discrete topology]] - the topology {{M|1=\mathcal{J}=\mathcal{P}(X)}} - where {{M|\mathcal{P}(X)}} denotes the [[Power set|power set]] of {{M|X}}
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*# [[Trivial topology]] - the topology {{M|1=\mathcal{J}=\{\emptyset, X\} }}
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==See Also==
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* [[Topology (subject)]]
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* [[Topological property theorems]]
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* [[Topology induced by a metric]]
  
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==References==
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<references/>
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{{Topology navbox|plain}}
 
{{Definition|Topology}}
 
{{Definition|Topology}}

Latest revision as of 13:37, 20 April 2016

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Definition

A topological space is a set [math]X[/math] coupled with a "topology", [ilmath]\mathcal{J} [/ilmath] on [math]X[/math]. We denote this by the ordered pair [ilmath](X,\mathcal{J})[/ilmath].

  • A topology, [ilmath]\mathcal{J} [/ilmath] is a collection of subsets of [ilmath]X[/ilmath], [math]\mathcal{J}\subseteq\mathcal{P}(X)[/math] with the following properties[1][2][3]:
  1. Both [math]\emptyset,X\in\mathcal{J}[/math]
  2. For the collection [math]\{U_\alpha\}_{\alpha\in I}\subseteq\mathcal{J}[/math] where [math]I[/math] is any indexing set, [math]\cup_{\alpha\in I}U_\alpha\in\mathcal{J}[/math] - that is it is closed under union (infinite, finite, whatever - "closed under arbitrary union")
  3. For the collection [math]\{U_i\}^n_{i=1}\subseteq\mathcal{J}[/math] (any finite collection of members of the topology) that [math]\cap^n_{i=1}U_i\in\mathcal{J}[/math]
  • We call the elements of [ilmath]\mathcal{J} [/ilmath] "open sets", that is [ilmath]\forall S\in\mathcal{J}[S\text{ is an open set}] [/ilmath], each [ilmath]S[/ilmath] is exactly what we call an 'open set'

As mentioned above we write the topological space as [math](X,\mathcal{J})[/math]; or just [math]X[/math] if the topology on [math]X[/math] is obvious from the context.

Comparing topologies

Given two topological spaces, [ilmath](X_1,\mathcal{J}_1)[/ilmath] and [ilmath](X_2,\mathcal{J}_2)[/ilmath] we may be able to compare them; we say:

Terminology If Comment
[ilmath]\mathcal{J}_1[/ilmath] coarser[2]/smaller/weaker [ilmath]\mathcal{J}_2[/ilmath] [ilmath]\mathcal{J}_1\subseteq\mathcal{J}_2[/ilmath] Using the implies-subset relation we see that [ilmath]\mathcal{J}_1\subseteq\mathcal{J}_2\iff\forall S\in\mathcal{J}_1[S\in\mathcal{J}_2][/ilmath]
[ilmath]\mathcal{J}_1[/ilmath] finer[2]/larger/stronger [ilmath]\mathcal{J}_2[/ilmath] [ilmath]\mathcal{J}_2\subseteq\mathcal{J}_1[/ilmath] Again, same idea, [ilmath]\mathcal{J}_2\subseteq\mathcal{J}_2\iff\forall S\in\mathcal{J}_2[S\in\mathcal{J}_1][/ilmath]
Grade: C
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The message provided is:
Need references for larger/smaller/stronger/weaker, Check Introduction To Topology - Mendelson, Addendum: investigate relating this to a poset (easy enough - not very useful / lacking practical applications)

Examples

See Also

References

  1. Topology - James R. Munkres
  2. 2.0 2.1 2.2 Introduction to Topological Manifolds - John M. Lee
  3. Introduction to Topology - Bert Mendelson