Second countable and sequentially compact topological spaces are compact. Proof
3708
How can I prove that compact subset iff sequentially compact in second countable topological spaces?
64694
$C_{c}(X)$ is complete. then implies that $X$ is compact.
148516
Proof that $\omega^\omega$ is completely metrizable and second countable
16649
Topological space with countable open cover $\{U_\alpha\} $ with each $U_\alpha$ second-countable, is second countable
71916
Quasi-components and components coincide for compact Hausdorff spaces
136094
$G_\delta$ singletons in compact Hausdorff and first countability
128730
Open subspaces of locally compact Hausdorff spaces are locally compact
150796
Subspace topology and order topology (2)
147331
Compactness and Hausdorffness
67259
Can we prove that a bounded closed subset of $\mathbb R^n$ is compact without Axiom of Choice?
154978
Some equivalent formulations of compactness of a metric space
29744
Compactness and axiomatisability
62793
$\mathbb{R}$ with the lower limit topology is not second-countable
7723
Countable basis of function spaces
65180
A compact Hausdorff space that is not metrizable
19566
Continuous images of compact sets are compact
94485
Limit point compactness implies compactness.