Covering Maps

• $E$,
  • toplogical sp.
• $X$,
  • toplogical sp.
• $\pi:E \rightarrow X$

Definition

$U$ is said to be evenly covered by $\pi$ if

$\pi^{-1}(U) \cong U$ by $\pi$.

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Definition

$\pi$ is said to be covering map if

• $E$ and $X$ are connected,
• $E$ and $X$ are locally path-connected
• $\pi$ is surjective
• $\pi$ is continuous
• $p \in X$ has a neighborhood $U$ that is evenly covered by $\pi$

In this case,

• $X$ is called the base of the covering
• $E$ is called a covering space of $X$,

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Reference