Prove that if $p: Y \to X$ is a covering space and $X$ is path connected, then the cardinality of $p^{-1}(X)$ is constant.
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 $x \in X$.
39455
Prove that $(X\times Y)\setminus (A\times B)$ is connected
72556
Is the closure of path-connected set path connected?
152518
Proof that the set of all possible curves is of cardinality $\aleph_2$?
1453
Prove that the graph is connected
13468
When does open and connected imply path-connected?
63442
Show that if $U$ is an open connected subspace of $\mathbb{R}^2$, then $U$ is path connected
152927
seemingly nontrivial question about covering maps and evenly covered open sets
35631
$H_0(X,A) = 0 \iff A \cap X_i \neq \emptyset \forall $ path-components $X_i$
66895
A condition for the image of a metrizable space to be metrizable
151051
connectedness of uniform topology
60981
Understanding relation between Quotient Space and $S^1$
102484
Connectedness problem and countability of compact subsets of the Sorgenfrey line
142631
Show that A has the same cardinality as $A\setminus\{x,y,z\}$
14345
If $A\subset\mathbb{R^2}$ is countable, is $\mathbb{R^2}\setminus A$ path connected?
24767
$N(X)$ denotes collection of all nets of $X$. $T:N(X)\to \mathcal{P}(X)$ be a map such that $T\{x_\delta\}=T\{y_\gamma\}$ if one is subnet of other.
48557
Product of connected spaces