---- > [!theorem] Theorem. ([[homeomorphic topological spaces have isomorphic fundamental groups]]) > Let $X,Y$ be [[topological space|topological spaces]]. If $h:(X,x_{0}) \to (Y,y_{0})$, $y_{0}=h(x_{0})$, is a [[homeomorphism]] of $X$ with $Y$, then the [[homomorphism of fundamental groups induced by a continuous map|induced homomorphism]] $h_{*}$ is an [[group isomorphism|isomorphism]] of [[fundamental group|fundamental groups]] $\pi_{1}(X,x_{0})$ and $\pi_{1}(Y,y_{0})$. > [!proof]- Proof. ([[homeomorphic topological spaces have isomorphic fundamental groups]]) > Because $h$ is a [[homeomorphism]], its inverse $k=h^{-1}$ exists and is [[continuous]]. We claim that $k_{*}:\pi_{1}(Y,y_{0})\to\pi_{1}(X,x_{0})$ is the inverse of $h_{*}$. Indeed, using the functorial properties of the [[homomorphism of fundamental groups induced by a continuous map]], we have $\begin{align} (h_{*} \circ k_{*})= & (h \circ k)_{*}=i_{*_{X}}=\text{identity map.} \end{align}$ Likewise $(k_{*} \circ h_{*})=(k \circ h)_{*}=i_{*_{Y}}=\text{identity map}.$ Thus $h_{*}$ is [[bijection|bijective]] with inverse $k_{*}:=h_{*}^{-1}$. ---- #### ----- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM outgoing([[]]) FLATTEN file.tags GROUP BY file.tags as Tag