Homeomorphic topological spaces have isomorphic fundamental groups/Statement

Statement

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

• $\pi_1(X,p)\cong\pi_1(Y,\varphi(p))$ - where $\cong$ denotes group isomorphism here, but can also be used to denote topological isomorphism (AKA: homeomorphism)

That is to say:

• $\big(X\cong_\varphi Y)\implies(\pi_1(X,p)\cong_{\varphi_*}\pi_1(Y,\varphi(p))\big)$