Hint request: Proving that every nonempty perfect set in $\mathbb{R}^{k}$ is uncountable
114770
Proof that a perfect set is uncountable
132382
Showing a set is nowhere dense in $C([0,1])$
73350
Geometric solution to find a Set of totally ordered Subsets of $ \mathbb{N} $ is uncountable
150605
Existence of a regular uncountable $\aleph_{\alpha}$ without $\mathsf{AC}$
124847
Is the set of surjective functions from $\mathbb{N}$ to $\mathbb{N}$ uncountable?
3217
Construction of uncountably many non-isomorphic linear (total) orderings of natural numbers
39351
$f(X)$ is uncountable and hence $X$ is uncountable.
92639
Prove that the family of open sets in $\mathbb{R}$ has cardinality equal to $2^{\aleph_0}$
157513
How can I prove that these sets are countable infinite?
41192
Neatest proof that set of finite subsets is countable?
78117
Closed property of nonempty finite set
99854
Must an uncountable subset of R have uncountably many accumulation points?
138202
Every open set in $\mathbb{R}$ is a countable union of closed sets
127678
A partition of the unit interval into uncountably many dense uncountable subsets
48523
Let $A$ be any uncountable set, and let $B$ be a countable subset of $A$. Prove that the cardinality of $A = A - B $
118595
Proving that a set is countable by finding a bijection
86685
Example of a set $S$ that is countable, but the set of limit points is uncountable