Difference between revisions of "Topological space"
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==[[Topological space/Definition|Definition]]== | ==[[Topological space/Definition|Definition]]== | ||
{{:Topological space/Definition}} | {{:Topological space/Definition}} | ||
| − | + | ==Comparing topologies== | |
| + | 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: | ||
| + | {| class="wikitable" border="1" | ||
| + | |- | ||
| + | ! Terminology | ||
| + | ! If | ||
| + | ! Comment | ||
| + | |- | ||
| + | ! {{Anchor|Coarser|Smaller|Weaker}}{{M|\mathcal{J}_1}} coarser{{rITTMJML}}/smaller/weaker {{M|\mathcal{J}_2}} | ||
| + | | {{M|\mathcal{J}_1\subseteq\mathcal{J}_2}} | ||
| + | | 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]}} | ||
| + | |- | ||
| + | ! {{Anchor|Finer|Larger|Stronger}}{{M|\mathcal{J}_1}} finer{{rITTMJML}}/larger/stronger {{M|\mathcal{J}_2}} | ||
| + | | {{M|\mathcal{J}_2\subseteq\mathcal{J}_1}} | ||
| + | | Again, same idea, {{M|\mathcal{J}_2\subseteq\mathcal{J}_2\iff\forall S\in\mathcal{J}_2[S\in\mathcal{J}_1]}} | ||
| + | |} | ||
| + | {{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)}} | ||
==Examples== | ==Examples== | ||
* Every [[Metric space|metric space]] induces a topology, see [[Topology induced by a metric|the topology induced by a metric space]] | * 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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*# [[Trivial topology]] - the topology {{M|1=\mathcal{J}=\{\emptyset, X\} }} | *# [[Trivial topology]] - the topology {{M|1=\mathcal{J}=\{\emptyset, X\} }} | ||
==See Also== | ==See Also== | ||
| − | * [[Topology]] | + | * [[Topology (subject)]] |
* [[Topological property theorems]] | * [[Topological property theorems]] | ||
| + | * [[Topology induced by a metric]] | ||
==References== | ==References== | ||
<references/> | <references/> | ||
| − | + | {{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]:
- Both [math]\emptyset,X\in\mathcal{J}[/math]
- 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")
- 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] |
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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
- Every metric space induces a topology, see the topology induced by a metric space
- Given any set [ilmath]X[/ilmath] we can always define the following two topologies on it:
- Discrete topology - the topology [ilmath]\mathcal{J}=\mathcal{P}(X)[/ilmath] - where [ilmath]\mathcal{P}(X)[/ilmath] denotes the power set of [ilmath]X[/ilmath]
- Trivial topology - the topology [ilmath]\mathcal{J}=\{\emptyset, X\}[/ilmath]
See Also
References
- ↑ Topology - James R. Munkres
- ↑ 2.0 2.1 2.2 Introduction to Topological Manifolds - John M. Lee
- ↑ Introduction to Topology - Bert Mendelson
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