Homeomorphic topological spaces have isomorphic fundamental groups/Statement

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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]


  1. Introduction to Topological Manifolds - John M. Lee