Difference between revisions of "Given a homeomorphism all subspaces of the domain are homeomorphic to their image under the homeomorphism itself"

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(Created page with "{{Stub page|grade=A**|msg=Come back and check this}} __TOC__ ==Statement== Let {{Top.|X|J}} and {{Top.|Y|K}} be topological spaces and suppose that {{M|f:X\rightarrow Y}}...")
 
(Factored statement out - with slight edit)
 
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__TOC__
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==Statement==
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=={{subpage|Statement}}==
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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:
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{{/Statement}}
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* {{M|\forall A\in\mathcal{P}(X)[A\cong_{f\vert_{A}^{\text{Im} } } f(A)]}}
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** In words: For all [[topological subspace|subspaces]] of {{M|X}} {{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}}
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==Proof==
 
==Proof==
 
{{Requires proof|grade=E|msg=I've done the proof on paper here:
 
{{Requires proof|grade=E|msg=I've done the proof on paper here:

Latest revision as of 13:25, 7 March 2017

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Statement

Let (X,J) and (Y,K) be topological spaces and suppose that f:X→Y is a homeomorphism between them, so X≅fY, then:

  • ∀A∈P(X)[A≅f|ImAf(A)]
    • In words: For all subspaces of X, suppose in particular A is a subspace, then f|ImA:A→f(A) - the restriction onto its image of f to A - is a homeomorphism between A and f(A)⊆Y
      • So A≅f|ImAf(A) explicitly

Proof

Grade: E
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I've done the proof on paper here:

  • Paper proof

It uses A⊆B⟹f−1(A)⊆f−1(B) which is somewhere under function properties, and it also expresses:

  • (f|ImA)−1(U)=f−1(U)∩A
which is ripe for factoring out Alec (talk) 13:31, 20 February 2017 (UTC)

References

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