[증명] open subspace of evenly covered space is evenly covered.

Proposition.

 

Let $p:E\rightarrow B$ a covering map. If $U\subset B$ is evenly covered by p and $U'\subset U$ is open,  $U'$ is evenly covered by p.

Proof.

Let $\{V_\alpha\}$ be the partition of $p^{-1}(U)$ into pieces and $p|_{V_\alpha}:V_\alpha\rightarrow U$, a homeomorphism. Let $V_\alpha=(p|_{V_\alpha})^{-1}(U)$ are disjoint, so $(p|_{V_\alpha})^{-1}(U')$ are disjoint and $\bigcup (p|_{V_\alpha})^{-1}(U')=p^{-1}(U')$. Let $W_\alpha=(p|_{V_\alpha})^{-1}(U')$. $p|_{W_\alpha}:W_\alpha\rightarrow U'$ is a restriction of $p|_{V_\alpha}$, which is a homeomorphism, so it is a homeomorphism. This concludes $U'$ is evenly covered by $p$, and $\{W_\alpha\}$ is the partition of $p^{-1}(U')$ into pieces.

End of proof.

 

 

 

+

 

예전에는 챗 지피티한테 위상수학 물어보면 정신 못차렸는데 이제는 이런 간단한 증명 정도는 구글도 해준다..