Difference between revisions of "Subspace topology"
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(Created page with " ==Definition== We define the subspace topology as follows. Given a topological space <math>(X,\mathcal{J})</math> and any <math>Y\subset X</math> we ca...") |
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We may say "<math>Y</math> is a subspace of <math>X</math> (or indeed <math>(X,\mathcal{J})</math>" to implicitly mean this topology. | We may say "<math>Y</math> is a subspace of <math>X</math> (or indeed <math>(X,\mathcal{J})</math>" to implicitly mean this topology. | ||
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| + | ==Closed subspace== | ||
| + | If {{m|Y}} is a "closed subspace" of {{m|(X,\mathcal{J})}} then it means that {{M|Y}} is [[Closed set|closed]] in {{M|X}} and should be considered with the subspace topology. | ||
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| + | ==Open subspace== | ||
| + | {{Todo|same as closed, but with the word "open"}} | ||
| + | |||
| + | ==Open sets in open subspaces are open== | ||
| + | {{Todo|easy}} | ||
{{Definition|Topology}} | {{Definition|Topology}} | ||
Revision as of 05:07, 15 February 2015
Definition
We define the subspace topology as follows.
Given a topological space [math](X,\mathcal{J})[/math] and any [math]Y\subset X[/math] we can define a topology on [math]Y,\ (Y,\mathcal{J}_Y)[/math] where [math]\mathcal{J}_Y=\{Y\cap U|U\in\mathcal{J}\}[/math]
We may say "[math]Y[/math] is a subspace of [math]X[/math] (or indeed [math](X,\mathcal{J})[/math]" to implicitly mean this topology.
Closed subspace
If [ilmath]Y[/ilmath] is a "closed subspace" of [ilmath](X,\mathcal{J})[/ilmath] then it means that [ilmath]Y[/ilmath] is closed in [ilmath]X[/ilmath] and should be considered with the subspace topology.
Open subspace
TODO: same as closed, but with the word "open"
Open sets in open subspaces are open
TODO: easy
