Difference between revisions of "Topological space"
From Maths
m |
m |
||
| Line 1: | Line 1: | ||
==Definition== | ==Definition== | ||
| − | 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: | + | 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<ref name="Top">Topology - James R. Munkres - Second Edition</ref><ref name="ITTM">Introduction to Topological Manifolds - Second Edition - John M. Lee</ref><ref name="ITT">Introduction to Topology - Third Edition - Bert Mendelson</ref>: |
# Both <math>\emptyset,X\in\mathcal{J}</math> | # Both <math>\emptyset,X\in\mathcal{J}</math> | ||
| − | # For the collection <math>\{U_\alpha\}_{\alpha\in I}\ | + | # 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) |
| − | # For the collection <math>\{U_i\}^n_{i=1}\ | + | # 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 write the topological space as <math>(X,\mathcal{J})</math> or just <math>X</math> if the topology on <math>X</math> is obvious. | 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. | ||
| + | * We call the elements of {{M|\mathcal{J} }} "[[Open set|open sets]]" | ||
| − | + | ==Examples== | |
| − | + | * Every [[Metric space|metric space]] induces a topology, see [[Topology induced by a metric|the topology induced by a metric space]] | |
| + | * Given any set {{M|X}} we can always define the following two topologies on it: | ||
| + | *# [[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}} | ||
| + | *# [[Trivial topology]] - the topology {{M|1=\mathcal{J}=\{\emptyset, X\} }} | ||
==See Also== | ==See Also== | ||
* [[Topology]] | * [[Topology]] | ||
| + | * [[Topological property theorems]] | ||
==References== | ==References== | ||
| − | + | <references/> | |
{{Definition|Topology}} | {{Definition|Topology}} | ||
Revision as of 16:24, 14 August 2015
Contents
Definition
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[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)
- 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 write the topological space as [math](X,\mathcal{J})[/math] or just [math]X[/math] if the topology on [math]X[/math] is obvious.
- We call the elements of [ilmath]\mathcal{J} [/ilmath] "open sets"
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]
