If X has a countable base, then every base contains a countable base.
33207
Bases having countable subfamilies which are bases in second countable space
147136
Subset of a countable set is itself countable
44974
Infinite set and countable subsets
59885
Proof Verification - Every sequence in $\Bbb R$ contains a monotone sub-sequence
113534
Show that any metrizable space $X$ is Hausdorff
78225
Countable additivity implies countable subadditivity?
31509
Is axiom of choice necessary for proving that every infinite set has a countably infinite subset?
138039
If $X$ is Hausdorff then every finite set is closed.
17624
Axiom of Choice iff Every set has a choice function
67050
Every separable metric space has a countable base
92249
Every finite set contains its supremum: proof improvement.
110635
Proving $\mathbb{N}^k$ is countable
68230
The cardinality of a countable union of sets with less than continuum cardinality
66398
Union of a finite set and a countably infinite set is countably infinite
157513
How can I prove that these sets are countable infinite?
11817
Proving that every set $A \subset \Bbb N$ of size $n$ contains a subset $B \subset A$ with $n | \sum_{b \in B} b$
48523
Let $A$ be any uncountable set, and let $B$ be a countable subset of $A$. Prove that the cardinality of $A = A - B $