View source for Given a homeomorphism all subspaces of the domain are homeomorphic to their image under the homeomorphism itself/Statement

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==Statement==
</noinclude>Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]] and suppose that {{M|f:X\rightarrow Y}} is a [[homeomorphism]] between them, so {{M|X\cong_f Y}}, then:
* {{M|\forall A\in\mathcal{P}(X)[A\cong_{f\vert_{A}^{\text{Im} } } f(A)]}}
** In words: For all [[topological subspace|subspaces]] of {{M|X}}, suppose in particular {{M|A}} is a subspace, then {{M|f\vert_{A}^\text{Im}:A\rightarrow f(A)}} - the [[restriction onto its image]] of {{M|f}} to {{M|A}} - is a homeomorphism between {{M|A}} and {{M|f(A)\subseteq Y}}
*** So {{M|A\cong_{f\vert_A^\text{Im} }f(A)}} explicitly<noinclude>
==Notes==
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==References==
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