X a locally compact Haussdorf space, each singleton of which is the intersection of countably many open sets. Show that X is first countable.
136094
$G_\delta$ singletons in compact Hausdorff and first countability
33157
Proof that an infinite subset of $\mathbb{N}$ is countable
67050
Every separable metric space has a countable base
93419
Let $\mathcal A$ be a collection of pairwise disjoint subsets of a $\sigma$-finite measure space,Show that $\mathcal A$ is at most countable.
149980
Are there any countable Hausdorff connected spaces?
17842
Is every set in a separable metric space the union of a perfect set and a set that is at most countable?
84460
If a set is $F_\sigma$ and is of first category then it is a countable union of closed nowhere dense sets
143484
A separable locally compact metric space is compact iff all of its homeomorphic metric spaces are bounded
62793
$\mathbb{R}$ with the lower limit topology is not second-countable
106606
How to prove that compact subspaces of the Sorgenfrey line are countable?
88904
Help with a proof. Countable sets.
120135
Show every nonempty compact Hausdorff space is not the countable union of nowhere dense sets
19297
Is the cofinite topology on an uncountable set first countable?
7678
Show "countable complement topology" is a topology
70048
In which topologies do open sets maintain open under countable or arbitrary intersection?
16649
Topological space with countable open cover $\{U_\alpha\} $ with each $U_\alpha$ second-countable, is second countable
80414
Countable Product of Sequentially Compact spaces is Sequentially Compact