Section 53: Covering Spaces

• $\{U_{\alpha}\}$ is the partition of $p^{-1}(V)$ into slices.
• If $V$ is connected, then such partitioning is unique.

• If $p$ is a covering map, and $p^{-1}(b)$ has $k$ elements for every $b\in B$ , then $E$ is called a k-fold covering of $B$ .
  • If $B$ is connected and $p^{-1}(b)$ has $k$ elements for some$b$ , then it has $k$ elements for every$b$ .
• If $p$ is a covering map, then $p$ is an open map.
• If $p$ is a covering map, then $p^{-1}(b)$ is discrete for each $b\in B$ .
• If $p$ is a covering map, then $p$ is a local homeomorphism, but not vice versa.
  • In general, the restriction of a covering map onto a subdomain is not a covering map, but may still be a local homeomorphism.
• $p|_{p^{-1}(B_{0})}$ where $B_{0}\subseteq B$ is a covering map.
• $p\times p'$ is a covering map.
• $p'\circ p$ is a covering map if $p'^{-1}(c)$ is finite for each $c$ .
• If $B$ is one of Hausdorff, regular, completely regular, or locally compact Hausdorff, then so is $E$ . If $B$ is compact and $p^{-1}(b)$ is always finite, $E$ is compact.