View source for Subspace topology

{{Refactor notice|grade=A|msg=Needed urgently, ready to plough on with it now though!}}
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==Definition==
Let {{Top.|X|J}} be a [[topological space]] and let {{M|S\in\mathcal{P}(X)}}<ref group="Note">Recall {{M|\mathcal{P}(X)}} denotes the [[power set]] of {{M|X}} and {{M|S\in\mathcal{P}(X)\iff S\subseteq X}}, so it's another way of saying let {{M|S}} be a subset of {{M|X}}, possibly empty, possibly equal to {{M|X}} itself</ref> be given. We can construct a new topological space, {{M|(S,\mathcal{J}_S)}} where the [[topology]] {{M|\mathcal{J}_S}} is known as {{nowrap|"the ''subspace topology''}} on {{M|S}}"{{rITTMJML}} ({{AKA}}: ''relative topology'' on {{M|S}}<ref name="ITTMJML"/>) and is defined as follows:
* {{M|1=\mathcal{J}_S:=\{U\cap S\ \vert\ U\in\mathcal{J}\} }} - the open sets of {{M|(S,\mathcal{J}_S)}} are precisely the intersection of open sets of {{Top.|X|J}} with {{M|S}} 
** '''Claim 1: ''' this is indeed a [[topology]]<ref name="ITTMJML"/>
Alternatively:
* '''Claim 2: ''' {{M|1=\forall U\in\mathcal{P}(S)\big[U\in\mathcal{J}_S\iff\exists V\in\mathcal{J}[U=S\cap V]\big]}}<ref name="ITTMJML"/>
We get with this a map, called the ''[[canonical injection of the subspace topology]]'', often denoted {{M|i_S:S\rightarrow X}} or {{M|\iota_S:S\rightarrow X}} given by {{M|i_S:s\mapsto s}}. This is an example of an [[inclusion map]], and it is [[continuous]].

Note that if one proves {{M|i_S}} is continuous then the [[characteristic property of the subspace topology|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 {{m|i_S}} comes from it as a [[corollary]]

==Terminology==
* Let {{M|U\in\mathcal{P}(S)}} be given. For clarity rather than saying {{M|U}} is [[open set|open]], or {{M|U}} is [[closed set|closed]] (which is surprisingly ambiguous when using subspaces) we instead say:
*# {{M|U}} is ''[[relatively open]]''<ref name="ITTMJML"/> - indicating we mean open in the subspace, or
*# {{M|U}} is ''[[relatively closed]]''<ref name="ITTMJML"/> - indicating we mean closed in the subspace
{{Todo|Closed and open subspace terminology, For example if {{M|S\in\mathcal{P}(X)}} is ''closed'' with respect to the topology {{M|\mathcal{J} }} on {{M|X}}, then we call {{M|S}} imbued with the subspace topology a ''closed subspace''}}
==[[Characteristic property of the subspace topology|Characteristic property]]==
{{:Characteristic property of the subspace topology/Statement}}
==Proof of claims==
===Claim 1: {{M|\mathcal{J}_S}} is a [[topology]]===
{{Requires proof|grade=C|msg=Really easy, hence low importance|easy=true}}
===Claim 2: Equivalent formulation of the relatively open sets===
{{Requires proof|grade=C|msg=Really easy, hence low importance|easy=true}}
==See next==
{{Todo|Theorems and propositions involving subspaces}}
==See also==
* [[Topological embedding]]
{{Todo|Link to more things}}
==Notes==
<references group="Note"/>
==References==
<references/>
{{Topology navbox|plain}}
{{Definition|Topology}}

=OLD PAGE=
==Definition==
Given a [[Topological space|topological space]] {{M|(X,\mathcal{J})}} and given a {{M|Y\subset X}} ({{M|Y}} is a subset of {{M|X}}) we define the ''subspace topology'' as follows:<ref name="Topology">Topology - Second Edition - Munkres</ref>
* {{M|(Y,\mathcal{K})}} is a topological space where the [[Open set|open sets]], {{M|\mathcal{K} }}, are given by {{M|1=\mathcal{K}:=\{Y\cap V\vert\ V\in\mathcal{J}\} }}
We may say any one of:
# Let {{M|Y}} be a subspace of {{M|X}}
# Let {{M|Y}} be a subspace of {{M|(X,\mathcal{J})}} 
and it is taken implicitly to mean {{M|Y}} is considered as a topological space with the ''subspace topology'' inherited from {{M|(X,\mathcal{J})}}

==Proof of claims==
{{Begin Theorem}}
Claim 1: The subspace topology is indeed a topology
{{Begin Proof}}
Here {{M|(X,\mathcal{J})}} is a topological space and {{M|Y\subset X}} and {{M|\mathcal{K} }} is defined as above, we will prove that {{M|(Y,\mathcal{K})}} is a topology.


Recall that to be a topology {{M|(Y,\mathcal{K})}} must have the following properties:
# {{M|\emptyset\in\mathcal{K} }} and {{M|Y\in\mathcal{K} }}
# For any {{M|U,V\in\mathcal{K} }} we must have {{M|U\cap V\in\mathcal{K} }}
# For an [[Indexing set|arbitrary family]] {{M|\{U_\alpha\}_{\alpha\in I} }} of open sets (that is {{M|\forall\alpha\in I[U_\alpha\in\mathcal{K}]}}) we have:
#* {{MM|\bigcup_{\alpha\in I}A_\alpha\in\mathcal{K} }}


'''Proof:'''
# First we must show that {{M|\emptyset,Y\in\mathcal{K} }}
#: Recall that {{M|\emptyset,X\in\mathcal{J} }} and notice that:
#:* {{M|1=\emptyset\cap Y=\emptyset}}, so by the definition of {{M|\mathcal{K} }} we have {{M|\emptyset\in\mathcal{K} }}
#:* {{M|1=X\cap Y=Y}}, so by the definition of {{M|\mathcal{K} }} we have {{M|Y\in\mathcal{K} }}
# Next we must show...

{{Todo|Easy work just takes time to write!}}
{{End Proof}}{{End Theorem}}

==Terminology==
* A '''closed subspace''' (of {{M|X}}) is a subset of {{M|X}} which is closed in {{M|X}} and is imbued with the subspace topology
* A '''open subspace''' (of {{M|X}}) is a subset of {{M|X}} which is open in {{M|X}} and is imbued with the subspace topology
{{Todo|Find reference}}
* A set {{M|U\subseteq X}} is '''open relative to {{M|Y}}''' (or [[Relatively open|''relatively open'']] if it is obvious we are talking about a subspace {{M|Y}} of {{M|X}}) if {{M|U}} is open in {{M|Y}}
** This implies that {{M|U\subseteq Y}}<ref name="Topology"/>
* A set {{M|U\subseteq X}} is '''closed relative to {{M|Y}}''' (or [[Relatively closed|''relatively closed'']] if it is obvious we are talking about a subspace {{M|Y}} of {{M|X}}) if {{M|U}} is [[Closed set|closed]] in {{M|Y}}
** This also implies that {{M|U\subseteq Y}}

==Immediate theorems==
{{Begin Theorem}}
Theorem: Let {{M|Y}} be a subspace of {{M|X}}, if {{M|U}} is open in {{M|Y}} and {{M|Y}} is open in {{M|X}} then {{M|U}} is open in {{M|X}}<ref name="Topology"/>
{{Begin Proof}}
This may be easier to read symbolically:
* if {{M|U\in\mathcal{K} }} and {{M|Y\in\mathcal{J} }} then {{M|U\in\mathcal{J} }}


'''Proof:'''
: Since {{M|U}} is open in {{M|Y}} we know that:
:* {{M|1=U=Y\cap V}} for some {{M|V}} open in {{M|X}} (for some {{M|V\in\mathcal{J} }})
: Since {{M|Y}} and {{M|V}} are both open in {{M|X}} we know:
:* {{M|Y\cap V}} is open in {{M|X}}
: it follows that {{M|U}} is open in {{M|X}}
{{End Proof}}{{End Theorem}}

==References==
<references/>

{{Definition|Topology}}