Does Compact Imply Separable?

We also have the following easy fact: Proposition 2.3 Every totally bounded metric space (and in particular every compact met- ric space) is separable. Intuitively, a separable space is one that is “well approximated by a countable subset”, while a compact space is one that is “well approximated by a finite subset”.

Does compact imply second countable?

Theorem 1. Every compact metrizable space is second-countable. ... Let X be a compact metrizable space, and let d be a metric on X that induces the topology on X. For each n ∈ Z+ let An be an open covering of X with 1/n-balls.

Is a subspace of a separable space separable?

2: A subspace of a separable metric space is separable.