Difference between revisions of "Subspace topology"

Revision as of 05:07, 15 February 2015

We define the subspace topology as follows.

Given a topological space

$(X,\mathcal{J})$ and any

$Y\subset X$ we can define a topology on

$Y,\ (Y,\mathcal{J}_Y)$ where

$\mathcal{J}_Y=\{Y\cap U|U\in\mathcal{J}\}$

We may say "

$Y$ is a subspace of

$X$ (or indeed

$(X,\mathcal{J})$ " to implicitly mean this topology.

If $Y$ is a "closed subspace" of $(X,\mathcal{J})$ then it means that $Y$ is closed in $X$ and should be considered with the subspace topology.

TODO: same as closed, but with the word "open"

TODO: easy

@@ Line 6: / Line 6: @@
 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 {{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.
+==Open subspace==
+{{Todo|same as closed, but with the word "open"}}
+==Open sets in open subspaces are open==
+{{Todo|easy}}
 {{Definition|Topology}}