Section 30: The Countability Axioms

1. First countability axiom: for every point $x\in X$ there is a countable basis at $x$ . $X$ is called first-countable.
2. Continuous functions and converging sequences in first-countable spaces (compare to §21):
3. Converging sequences of points and the closure (The Sequence Lemma): let $X$  be a topological space, then
4. in any topological space: if there is a sequence of points of $A\subseteq X$ converging to $x$ then $x\in\overline{A}$ .
5. if $X$ is first-countable: if $x\in\overline{A}$ then there is a sequence of points of $A$ converging to $x$ .
6. Continuity and convergent sequences of points: let $f:X\rightarrow Y$ , then
7. in any topological space: if $f$ is continuous at $x$ then for every $x_n\rightarrow x$ : $f(x_n)\rightarrow f(x)$ .
8. if $X$ is first-countable (and $Y$ is an arbitrary space): if for every $x_n\rightarrow x$ : $f(x_n)\rightarrow f(x)$ then $f$ is continuous at $x$ .
9. Lindelöf space: every open covering has a countable subcovering.
10. Separable space: there is a countable dense. A dense is a subset of a space such that its closure is the whole space (every point not in it is its limit point).
11. Second countability axiom: $X$ has a countable basis for its topology. $X$ is said to be second-countable.
12. A second countable space is both Lindelöf and separable.
13. If a metric space is Lindelöf or separable then it is second countable.
14. Subspaces and countability axioms:
15. A subspace of a first-countable (second-countable) space is first-countable (second-countable).
16. A subspace of a separable (Lindelöf) space need NOT be a separable (Lindelöf) space.
17. An open subspace of a separable space is separable.
18. A closed subspace of a Lindelöf space is Lindelöf.
19. If a subspace of a second-countable space is discrete then it is countable.
20. If a subset of a second-countable space is uncountable then only countably many points in the subset are not its limit points.
21. Products and countability axioms:
22. A countable product of separable (first-countable, second-countable) spaces (in the product topology) is separable (first-countable, second-countable).
23. A product of two Lindelöf spaces does NOT have to be Lindelöf.
24. Lindelöf$\times$ compact is Lindelöf.
25. A continuum product of separable spaces is separable.
26. If the product of any number of spaces possesses one of these 4 properties, so does each space in the product.
27. Continuous functions and countability axioms:
28. The continuous image of a first-countable space need NOT be first-countable.
29. The continuous (even the quotient) image of a second-countable space need NOT be second-countable.
30. The continuous open image of a first-countable (second-countable) space is first-countable (second-countable).
31. The continuous image of a separable (Lindelöf) space is separable (Lindelöf).
32. Topological groups and countability axioms:
33. If a topological group is first-countable and (separable or Lindelöf) then it is second-countable.