What are the Results of the First and Second Axioms of Countability?

Question

What are the consequences of a space being first or second countable? What was the motivation for these axioms in the first place?

Answer 1

From a point of view of a general topologist, these spaces are nice. For topologists which work with cardinal invariants, many of them collapse for these spaces. By Tichonoff-Urysohn Theorem, a second countable space is metrizable iff it is regular (in this case it is homeomorphic to a subspace of the Hilbert cube $[0,1]^\omega$). First countable spaces are rarely metrizable$^\star$, but, nevertheless, in these spaces a closure can be defined by convergent sequences. First (and second too) countable spaces are preserved by open continuous maps. In fact, by Ponomarev Theorem, each first-countable space of infinite cardinality is an open continuous image of a zero-dimensional metrizable space of the same weight.

$^\star$ But, by Birkhoff-Kakutani Theorem, a $T_0$ topological group is metrizable iff it is first countable.