π-base

This little section of my blog is dedicated to my own proofs of as many theorems in π-base as possible. Could be used as a reference to all the proofs in one single place, instead of relying on looking through the books their website is referencing. Keep in mind, the simpler proofs are mine — a.k.a likely less elegant than from other sources — and the harder proofs taken from other sources were adapted to fit my writing style.

The prerequisites to understand the proofs are included in the important information section, below. Additionally, the space is always assumed to be $X$. If it has a metric/pseudometric, it is assumed to be $d$. The letters $U,V,W$ always refer to open sets. Every instance of the word “neighborhood(s)” is shortened to “nbd(s)” and there is no harm to assume it’s open whenever needed.

Important: π-base is updating constantly and my website is not updating as fast, so some theorems may be missing, or in an outdated form. The last update was: Mar 11, 2026.

Current progress: 400 / 858 (46.62%)

Out of the 400 proofs, here’s a breakdown by difficulty:

You can list through my proofs by ID or by difficulty (a personal metric, of course).

By ID: 0xx, 1xx, 2xx, 3xx, 4xx, 5xx, 6xx, 7xx, 8xx.

By difficulty: 1, 2, 3, 4, 5, 6, 7, 8.

Alternatively, you can see the latest 50 proofs.

Important information

Prerequisites

Basic understanding of topological concepts is necessary. If I’m talking about a continuous function $f$, I may use the fact that $f(\cl{A}) \subseteq \cl{f(A)}$ or that $f^{-1}(\cl{A})$ is a closed set. Some level of knowledge about ordinals is important, because there are many definitions involving ordinals, such as radial spaces. In fact, the common notation for sequences is to denote $(x_n)_{n < \omega}$ instead of $\{x_n\}_{n=0}^\infty$. $n < \omega$ just means $n$ is a natural number (note that $0 \in \omega$ while some people adopt the notion that $0 \notin \N$, so you could argue $\omega \ne \N$).

Of course, to understand a proof, it’s important to understand the definitions. For example, a space that is exhaustible by compacts is, by definition, a space that is $\sigma$-compact and weakly locally compact. But this is equivalent to saying the space is a countable union of compact sets such that their interiors form an open cover. Whenever there are equivalent notions for a given space, I may freely choose one of the equivalent notions for my advantage in proofs.

On top of the basic concepts of topology, I will also make use of important results, which are sometimes the crux of a given proof. For example, I use Urysohn’s lemma to prove theorem 37. I classified that proof as a difficulty 3, but if I couldn’t use Urysohn’s lemma, it would’ve easily been a 6 or more. Conversely, I do try to use these results fairly. For example, theorem 23 is precisely known to be the $\implies$ direction of the Nagata-Smirnov metrization theorem. I could easily just say “Immediate from the Nagata-Smirnov metrization theorem” and classify as a difficulty 2 proof. I don’t think that’s fair, so instead, I decided to write a proof for that half of the theorem (which is definitely not something I could’ve come up with on my own, making it a very difficult proof). So here are some of the important results that I take as a given:

• Tube lemma

• Urysohn’s lemma

• Tietze extension theorem

• One-point compactification of a locally compact $T_2$ space is compact

Axiom of Choice

Every proof that uses the axiom of choice will have it mentioned explicitly. But the axiom of countable choice is essential to so many proofs, it is used implicitly. The implicit uses include:

• If $X$ is infinite, let $A \subseteq X$ be a countable infinite subset.

• For each $n$, choose an $x_n \in U_n$ such that…

• Each $A_n$ is countable, therefore, $\bigcup_{n < \omega} A_n$ is countable.