Covering Projections and Quotients

Question

Let $Y$ be a covering space of $X$, where $p:Y\to X$ is a covering projection. For $x\in X$, define the fiber of $x$ as $p^{-1}(x)$. Set up an equivalence relation on $Y$ as $y_1\sim y_2$ if they are in the same fiber. How do you show that $\sim$ is a closed relation? Further, how do you prove that $Y/\sim$ with the quotient topology is homeomorphic to $X$?

Answer 1

Every covering map $p:Y\to X$ is a local homeomorphism, meaning that every $y\in Y$ has a neighborhood $N$ such that the restriction $N\to p(N)$ is a homeomorphism and $p(N)$ is a neighborhood of $y$. Every local homeomorphism is an open map and thus a quotient map. This implies that the induced map $Y/{\sim} \to X$ is a quotient map and, since it is bijective, a homeomorphism.

Regarding the closedness of the relation $\sim$: One easily sees that the subset $\sim = \{(y,y') \mid y\sim y'\}$ of $Y\times Y$ is the preimage of the diagonal $\Delta_X = \{(x,x)\mid x\in X\}$ under the map $p\times p : Y\times Y \to X\times X$. Since $X$ is Hausdorff, this diagonal is closed, hence its preimage is closed as well.