If $X$ is a covering space of $Y$ with the covering map $p$ and $Y$ is connected, then $p^{-1}(y)$ have the same cardinality for every $y\in Y$.
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 $x \in X$.
92639
Prove that the family of open sets in $\mathbb{R}$ has cardinality equal to $2^{\aleph_0}$
158189
If $\{E_\alpha\}$ is connected, $\bigcap\limits_{\alpha\in A}E \neq \emptyset$, then $\bigcup\limits_{\alpha\in A}E$ is connected
124234
Prove that if $X$ and it's closure $\overline X$ are connected and if $X\subset Y \subset \overline X$, show that Y is also connected.
156606
Universal covering spaces of homotopy equivalent spaces are homotopy equivalent
157002
Connected compact covering space of torus is homeomorphic to $S^1\times S^1$
59459
Binary sequences and ${2}^{\mathbb{N}}$ have the same cardinality
15733
Show that the cardinality of $\mathbb N$ is the same as the cardinality of $\mathbb N \times \mathbb N$
89167
Cardinality of equivalence relations in $\mathbb{N}$
151051
connectedness of uniform topology
145873
Let $X$ and $Y$ be finite sets. Then the set $Y^X$ is finite and $\#(Y^X) = (\#Y)^{\#X}$.
92870
composition of covering maps
142267
If two sets have the same cardinality, then so do their power sets. Converse can't be answered?
31266
Proof of "the continuous image of a connected set is connected"
150379
Perfect Map $p:\ X\to Y$, $Y$ compact implies $X$ compact
149382
Prove that $(0,1) \subseteq\mathbb R$ and $(4,10) \subseteq\mathbb R$ have the same cardinality.
89264
Connectedness in the Quotient Space