# Finitely sheeted covering space of a compact space is is compact

Let $$p:X\to Y$$ be a finitely sheeted covering space. I want to show that $$X$$ is compact if $$Y$$ is. I have proven the following lemma.

Let $$U\subset X$$ be open containing $$p^{-1}(b)$$, then there exists an open subset $$O_b$$ such that $$p^{-1}(O_b)\subset U$$.

Now, let $$\mathcal{U}$$ be an open cover of $$X$$. I choose a finite evenly covered cover $$\mathcal{V}$$ of $$Y$$, which is possible since $$Y$$ is compact. Let $$y\in Y$$, then there exists an $$U_i\in\mathcal{U}$$ such that $$y\in p(U_i)$$. How can I go on and invoke the lemma?

Let $$\mathcal{U}$$ be an open cover of $$X$$. For each $$y \in Y$$ the finite set $$p^{-1}[\{y\}]$$ is covered by finitely many members of $$\mathcal{U}$$, let's call these $$\mathcal{U}_y \subseteq \mathcal{U}$$ and let $$O_y$$ be open in $$Y$$ such that $$p^{-1}[O_y] \subseteq \bigcup \mathcal{U}_y$$, by your lemma.
Then $$\{O_y: y \in Y\}$$ is an open cover of $$Y$$ so by compactness of $$Y$$ there are finitely many $$O_{y_1}, \ldots, O_{y_n}$$ that cover $$Y$$. Now note that $$\bigcup_{i=1}^n \mathcal{U}_{y_i}$$ is a finite (finite union of finite sets) subcover of $$\mathcal{U}$$:
$$X= p^{-1}[Y]=p^{-1}[\bigcup_{i=1}^n O_{y_i}] = \bigcup_{i=1}^n p^{-1}[O_{y_i}] \subseteq \bigcup_{i=1}^n \bigcup \mathcal{U}_{y_i}$$
Note that we only really use the property of the lemma (which is equivalent to $$p$$ being a closed map) and all fibres being compact, i.e. $$p$$ is a perfect map, and preimages of a compact space under a perfect map are compact, as this proof essentially shows.
Let $$\mathcal U$$ be an open cover of $$X.$$ Let $$\mathcal V$$ be the family of all open sets $$V\subset Y$$ such that there exists a finite subset $$\mathcal U_V\subset\mathcal U$$ with $$p^{-1}(V)\subset \bigcup_{U\in \mathcal U_V}U.$$ Use your lemma to show that $$\mathcal V$$ is a cover of $$Y.$$ A finite subcover $$\mathcal V'\subset\mathcal V$$ of $$Y$$ gives a finite subcover $$\bigcup_{V\in\mathcal V'}\mathcal U_{V}\subset \mathcal U$$ of $$X.$$