Subspace topology

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

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

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Open sets in open subspaces are open

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TODO: easy

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