Covering Space

Definition

Given a space $X$, a covering space of $X$ consists of a space $\tilde{X}$ and a map $p:\tilde{X}\to X$ satisfying the following condition:

For each point $x\in X$ there is an open neighborhood $U$ of $x$ in $X$ such that $p^{-1}(U)$ is a union of disjoint open sets each of which is mapped homeomorphically onto $U$ by $p$. Such a $U$ is called evenly covered

Example

Think $X$ as $S^1$ and $\tilde{X}$ as $\mathbb{R}$, then we can construct $p$ (this spiral shape is called a helix) we define $p:\mathbb{R}\to S^1$ with $p(s)=(\cos (2\pi s),\sin (2\pi s))$ each section on $S_1$ is mapped from multiple (infinitely many) sections on $\mathbb{R}$ Each map from a section on $R$ to section on $S^1$ is a homeomorphism

Define ${\tilde{\omega }}_n:I\to \mathbb{R}$ as $p({\tilde{\omega }}_n(t)={\omega }_n(t)$ , then it is a lift of loop ${\omega }_n$ a) For each path $f:I\to X$ starting at a point $x_0\in X$ and each ${\tilde{x}}_0\in p^{-1}(x_0)$, there is a unique lift $\tilde{f}:I\to \tilde{X}$ starting at ${\tilde{x}}_0$. b) For each homotopy $f_t:I\to X$ of paths starting at $x_0$ and each ${\tilde{x}}_0\in p^{-1}(x_0)$, there is a unique lifted homotopy ${\tilde{f}}_t:I\to \tilde{X}$ of paths starting at ${\tilde{x}}_0$.