The key here is recognizing that many proofs regarding convergence in metric spaces involve constructing sequences of nested open sets about a point. Sometimes these describe the topological structure near the point and other times not. In what follows we identify the local characteristic of topological space that makes these proofs work, and prove that sequences suffice to describe the topological structure of spaces with this characteristic. 8 First-countability is hereditary and preserved by continuous open onto maps. Proof Left to the reader.
Complete separability implies first countability.
Converse not always true. Converse valid on a countable underlying set. This is why sequences are sufficiently general to describe, fully, metric and metrizable spaces. A contains every limit of every convergent sequence of its own points. N j A for all j 1.