View source for Homeomorphic topological spaces have isomorphic fundamental groups

{{Stub page|grade=A|msg=Review and demote}}
__TOC__
==[[/Statement|Statement]]==
{{/Statement}}
==Proof==
The idea is to recall that the definition of a [[categorical isomorphism]] means that the map composed with its inverse is the identity of the codomain and the inverse composed with the map is the identity on the domain. 

Noting that a [[homeomorphism]] is an instance of this we see that both compositions are identity maps, which as we know from [[the induced fundamental group homomorphism of the identity map is the identity map of the fundamental group]] and using:
* [[The induced fundamental group homomorphism of a composition of continuous maps is the same as the composition of their induced homomorphisms]]

Shows that the induced maps are {{plural|categorical isomorphism|s}} too. 

Using:
* [[A map is a group isomorphism if and only if it satisfies the properties of a categorical isomorphism]] and the result follows.
{{Requires proof|grade=A|msg=Good to do formally}}
==References==
<references/>
{{Definition|Algebraic Topology|Topology|Homotopy Theory}}