Second-Countable Space is First-Countable/Proof 2

Theorem

Let $T = \struct {S, \tau}$ be a topological space which is second-countable.


Then $T$ is also first-countable.


Proof

By definition of second-countable space, there exists a countable analytic basis $\BB \subseteq \tau$.

Then each $x \in S$ has a countable local basis:

$\BB_x := \set {U \in \BB: x \in U}$

$\blacksquare$



Although this article appears correct, it's inelegant. There has to be a better way of doing it.
In particular: details
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Improve}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.


This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.