Motivation for the question:
While I can appreciate the significance of Theorem 30.1 below, and also the Urysohn metrization theorem, I can't seem to understand why either of them is rooted in these countability axioms, as in what their essence is.
Question: Could someone explain the essence of how having such countability axioms leads to such nice results?
Relevant Definitions
First countability: A space X is said to have a countable basis at $x$ if there is a countable collection $B$ of neighborhoods of $x$ such that each neighborhood of $x$ contains at least one of the elements of $B$. A space that has a countable basis at each of its points is said to satisfy the first countability axiom.
Second countability: If a space $X$ has a countable basis for its topology, then $X$ is said to satisfy the second countability axiom.
Theorem 30.1. (Chapter-4 Munkres's Topology) Let $X$ be a topological space.
(a) Let $A$ be a subset of $X$. If there is a sequence of points of $A$ converging to $x$, then $x \in \bar{A}$; the converse holds if $X$ is first-countable.
(b) Let $f: X \rightarrow Y$. If $f$ is contınuous, then for every convergent sequence $x_{n} \rightarrow x$ in $X$, the sequence $f\left(x_{n}\right)$ converges to $f(x)$. The converse holds if $X$ is first countable.
(Earlier on) Our goal is to prove the Urysohn metrization theorem, if a topological space $X$ satisfies certain countability axiom (the second) and a certain seperation axiom (the regularity axiom), then $X$ can be imbedded in a metric space and is this metrizable
I am looking for something around the lines of the discussions in these questions:
- What should be the intuition when working with compactness?
- Why is a topology made up of 'open' sets?
Edit: I found out that the intuition behind this question is discussed in Tai Danae Bradley's Categorical approach to topology chapter-3. So check that out if you want a resolution.
