----- > [!proposition] Proposition. ([[condition for restriction of covering maps]]) > Let $p: E \to B$ be a [[covering space|covering map]]. If $B_{0}$ is a [[subspace topology|subspace]] of $B$, and if $E_{0}=p ^{-1}(B_{0})$, then the map $p |_{E_{0}}: E_{0} \to B_{0}$ obtained by restricting $p$ is a [[covering space|covering map]]. > [!proof]- Proof. ([[condition for restriction of covering maps]]) > > Let $b_{0} \in B_{0}$ and fix a [[neighborhood]] $U$ in $B$ of $b_{0}$ that is [[evenly covered]] by $p$. Then $B_{0} \cap U$ is an open (in $B_{0}$) [[neighborhood]] of $b_{0}$. Write $p ^{-1}(U)=\bigsqcup_{\alpha}V_{\alpha}$ > where each $V_{\alpha}$ is [[homeomorphism]] to $U$ via $p$. Then using [[preimages and intersections commute]] we have $\begin{align} > p ^{-1}(B_{0} \cap U) = & p ^{-1} (B_{0}) \cap p ^{-1}(U) \\ > = & E_{0} \cap \bigsqcup V_{\alpha} \\ > = & \bigsqcup_{\alpha} E_{0} \cap V_{\alpha} > \end{align}$ > where each $E_{0} \cap V_{\alpha}$ is mapped [[homeomorphism|homeomorphically]] onto $B_{0} \cap U$ by $p$. ----- #### ---- #### 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