# Homeomorphic topological spaces have isomorphic fundamental groups

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Review and demote

## Statement

Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be homeomorphic topological spaces, let [ilmath]p\in X[/ilmath] be given (this will be the base point of the fundamental group [ilmath]\pi_1(X,p)[/ilmath]) and let [ilmath]\varphi:X\rightarrow Y[/ilmath] be that homeomorphism. Then: [1]:

That is to say:

• [ilmath]\big(X\cong_\varphi Y)\implies(\pi_1(X,p)\cong_{\varphi_*}\pi_1(Y,\varphi(p))\big)[/ilmath]

## 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:

Shows that the induced maps are categorical isomorphisms too.

Using: