If $p: \tilde{X} \rightarrow X$ is a covering space, then the cardinality of the set $p^{-1}(x)$ is locally 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 $x \in X$.
84510
If $f(\mathbb{C})\subset \mathbb{C}-[0,1]$ then $f$ is constant
92639
Prove that the family of open sets in $\mathbb{R}$ has cardinality equal to $2^{\aleph_0}$
46892
Inclusion of $L^p$ spaces
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 $\{0,1\}$.
109070
Question about the proof of the distributive property for sets
37183
Show that the set of all $x$ such that $x \in P$ and $x \notin Q$ exists
133504
Suppose $X$ is a metric space and $S \subseteq X.$ Then, $S^o=\{x \in X~|~dist(x,S^c)>0\}$.
106534
"Closed subsets of compact space are compact" with an alternative definition of covering
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.
48792
Prove $p_n(x) \rightarrow \sqrt{x}$ uniformly as $n \rightarrow \infty$
89712
For every set $A$, the empty set is a subset of $A$. The empty set is a set. Therefore, the empty set has a cardinality $\geq 1\ldots$
79568
Cardinality of the set $D$
109054
Given an infinite set of cardinality $X$, is there a chain in $\mathcal P(X)$ of the same cardinalty of $\mathcal P(X)$?
58285
Is it true that if $X$ is connected, then for every nonempty proper subset $A$ of $X$, we have $\mathbf{Bd} \ne \emptyset$
139537
Show that $x\subseteq y \Leftrightarrow \mathcal{P}(x)\subseteq\mathcal{P}(y)$.
29655
$L^p$ and $L^q$ space inclusion