Difference between revisions of "Topological space/Definition"
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==Definition== | ==Definition== | ||
</noinclude>A ''topological space'' is a set <math>X</math> coupled with a "topology", {{M|\mathcal{J} }} on <math>X</math>. We denote this by the [[ordered pair]] {{M|(X,\mathcal{J})}}. | </noinclude>A ''topological space'' is a set <math>X</math> coupled with a "topology", {{M|\mathcal{J} }} on <math>X</math>. We denote this by the [[ordered pair]] {{M|(X,\mathcal{J})}}. | ||
| − | * A topology, {{M|\mathcal{J} }} is a collection of subsets of {{M|X}}, <math>\mathcal{J}\ | + | * A topology, {{M|\mathcal{J} }} is a collection of subsets of {{M|X}}, <math>\mathcal{J}\subseteq\mathcal{P}(X)</math> with the following properties{{rTJRM}}{{rITTMJML}}{{rITTBM}}: |
# 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}\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_\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> | # 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 {{M|\mathcal{J} }} "[[Open set|open sets]]", that is {{M|\forall S\in\mathcal{J} }}, {{M|S}} is exactly what we call an 'open set' | + | * We call the elements of {{M|\mathcal{J} }} "[[Open set|open sets]]", that is {{M|\forall S\in\mathcal{J}[S\text{ is an open set}] }}, each {{M|S}} 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.<noinclude> | + | 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.<noinclude> |
==References== | ==References== | ||
Revision as of 11:44, 20 April 2016
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.
