# Subspace topology

Jump to: navigation, search
Grade: A
This page is currently being refactored (along with many others)
Please note that this does not mean the content is unreliable. It just means the page doesn't conform to the style of the site (usually due to age) or a better way of presenting the information has been discovered.
The message provided is:
Needed urgently, ready to plough on with it now though!

## Definition

Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space and let [ilmath]S\in\mathcal{P}(X)[/ilmath][Note 1] be given. We can construct a new topological space, [ilmath](S,\mathcal{J}_S)[/ilmath] where the topology [ilmath]\mathcal{J}_S[/ilmath] is known as "the subspace topology on [ilmath]S[/ilmath]" (AKA: relative topology on [ilmath]S[/ilmath]) and is defined as follows:

• [ilmath]\mathcal{J}_S:=\{U\cap S\ \vert\ U\in\mathcal{J}\}[/ilmath] - the open sets of [ilmath](S,\mathcal{J}_S)[/ilmath] are precisely the intersection of open sets of [ilmath](X,\mathcal{ J })[/ilmath] with [ilmath]S[/ilmath]

Alternatively:

• Claim 2: [ilmath]\forall U\in\mathcal{P}(S)\big[U\in\mathcal{J}_S\iff\exists V\in\mathcal{J}[U=S\cap V]\big][/ilmath]

We get with this a map, called the canonical injection of the subspace topology, often denoted [ilmath]i_S:S\rightarrow X[/ilmath] or [ilmath]\iota_S:S\rightarrow X[/ilmath] given by [ilmath]i_S:s\mapsto s[/ilmath]. This is an example of an inclusion map, and it is continuous.

Note that if one proves [ilmath]i_S[/ilmath] is continuous then the characteristic property boils down to little more than the composition of continuous maps is continuous, if one proves the characteristic property first, then continuity of [ilmath]i_S[/ilmath] comes from it as a corollary

## Terminology

• Let [ilmath]U\in\mathcal{P}(S)[/ilmath] be given. For clarity rather than saying [ilmath]U[/ilmath] is open, or [ilmath]U[/ilmath] is closed (which is surprisingly ambiguous when using subspaces) we instead say:
1. [ilmath]U[/ilmath] is relatively open - indicating we mean open in the subspace, or
2. [ilmath]U[/ilmath] is relatively closed - indicating we mean closed in the subspace

TODO: Closed and open subspace terminology, For example if [ilmath]S\in\mathcal{P}(X)[/ilmath] is closed with respect to the topology [ilmath]\mathcal{J} [/ilmath] on [ilmath]X[/ilmath], then we call [ilmath]S[/ilmath] imbued with the subspace topology a closed subspace

## Characteristic property

 Diagram [ilmath]\xymatrix{ Y \ar[r]^f \ar[dr]_{i_S\circ f} & S \ar@{^{(}->}[d]^{i_S}\\ & X}[/ilmath]
Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space and let [ilmath](S,\mathcal{J}_S)[/ilmath] be any subspace of [ilmath](X,\mathcal{ J })[/ilmath][Note 2]. The characteristic property of the subspace topology is that:
• Given any topological space [ilmath](Y,\mathcal{ K })[/ilmath] and any map [ilmath]f:Y\rightarrow S[/ilmath] we have:
• [ilmath](f:Y\rightarrow S [/ilmath] is continuous[ilmath])\iff(i_S\circ f:Y\rightarrow X [/ilmath] is continuous[ilmath])[/ilmath]

Where [ilmath]i_S:S\rightarrow X[/ilmath] given by [ilmath]i_S:s\mapsto s[/ilmath] is the canonical injection of the subspace topology (which is itself continuous)[Note 3]

## Proof of claims

### Claim 1: [ilmath]\mathcal{J}_S[/ilmath] is a topology

Grade: C
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:
Really easy, hence low importance

This proof has been marked as an page requiring an easy proof

### Claim 2: Equivalent formulation of the relatively open sets

Grade: C
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:
Really easy, hence low importance

This proof has been marked as an page requiring an easy proof

## See next

TODO: Theorems and propositions involving subspaces

## See also

TODO: Link to more things