Let $p: Y \to X$ be a covering space and $X$ is connected. I want to show that $\forall x \in X$ the cardinality of $p^{-x}$ is the same.
$\textbf{My Attempt:}$ Let us first fix a point $x_0 \in X$ such that $|p^{-1}(x_0)|=\kappa$.
Then let $H=\{x \in X \mid |p^{-1}(x)=\kappa| \}$
Then let $f: H \to \text{Cardinals}$ be defined as
$f(x)=|p^{-1}(x)|$. Then $f$ is a constant function on $H$.
How do I show that $H$ is open? $H$ is the inverse image of $\{\kappa\}$ but this is a singleton set from the collection of cardinals. I am not really sure what to do with that.
If I can show that $H$ is open, then I can prove by contradiction, using the assumption of connectedness, that all $x \in X$ are such that $|p^{-1}(x)|= \kappa$.
