Definition

Let $p: E \to B$ be a continuous Surjective map; an open set $U$ in $B$ is evenly covered (by $p$) if $p^{-1} (U)$ is a Disjoint union (in the sense of union of disjoint sets) of open sets $V_\alpha$$ in $E$, $\alpha \in J$:

$p^{-1} (U) = \dot{\cup}_{\alpha \in J} V_\alpha$

and each restriction

$p|V_\alpha: V_\alpha \to U$

is homeomorphic

$E$ is then called a covering space

Definition. A continuous surjective map $p: E \to B$ is a covering if each $b \in B$ has a nbhd. $U$ which is evenly covered.

Example: $p: \mathbb{R} \to S^1 \subset \mathbb{C}$, $p(t) = e^{2\pi t}$

Another example $let n\in \mathbb{N}$ fixed.

$p_n: S^1 \to S^1 \subset \mathbb{C}$

$z \mapsto z^n$

Remark. $p: E \to B$ is a covering. Then each fiber $p^{-1}(b)$, $b \in B$, has the Discrete topology.

When $p: E \to B$ is a covering, and $B$ is a Connected space, then every fiber has the same number of elemments

Path lifting lemma

Homotopy lifting lemma

Corollarly. Path-homotopic paths map to path-homotopic paths using Homotopy lifting lemma

This can be used to derive the Fundamental group of the Circle

covering of connected space has unique liftings (if it exists).

Good example: I think the Riemann surface is a covering of the complex plane minus the origin. See examples in this video! (see here)