If $p : E \to X$ is a covering map and $E$ is path connected, then all the fibers have the same cardinality.
7665
If $h : Y \to X$ is a covering map and $Y$ is connected, then the cardinality of the fiber $h^{-1}(x)$ is independent of $x \in X$.
89264
Connectedness in the Quotient Space
102598
Show that $X$ is path connected if and only if each $X_\alpha$ is path connected.
23125
Prove that there is a bijection between the set of all subsets of $X$, $P(X)$, and the set of functions from $X$ to $\{0,1\}$.
146789
A locally constant sheaf on a locally connected space is a covering space; Proof?
57122
If $E$ is a connected space, then so is $\overline{E}$.
136158
$X$ has Indiscrete topology $\implies$ any subset of $X$ is connected
147197
Number of connected components of the complement of a compact
91163
Let the set S be infinite, and the set T countably infinite. Show that S and S U T have the same cardinality
102484
Connectedness problem and countability of compact subsets of the Sorgenfrey line
2575
Showing that $\mathbb{R}$ is connected
72556
Is the closure of path-connected set path connected?
9082
Do sets, whose power sets have the same cardinality, have the same cardinality?
96781
connected manifolds are path connected
144383
The closure of a connected set in a topological space is connected
55484
Proving that the map from a topological space to a fiber bundle is open.
13066
A Zariski open subset of a variety has the same dimension as the variety.