---
date: '2025-11-01'
description: separation axioms, urysohn lemma, metrization checkpoints for mit 18.901 week 4–5.
id: topology-separation
modified: 2026-06-05 15:08:21 GMT-04:00
seealso:
  - '[[thoughts/topology/point set|point-set]]'
  - '[[thoughts/topology/compactness|compactness]]'
  - '[[thoughts/topology|topology hub]]'
tags:
  - math
  - math/topology
title: separation axioms
created: '2025-11-01'
published: '2025-11-01'
pageLayout: default
slug: thoughts/topology/separation
permalink: https://aarnphm.xyz/thoughts/topology/separation.md
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---
## scope

- review $t_0$ through $t_4$ axioms with canonical examples and counterexamples.
- document urysohn lemma, tietze extension theorem, stone–cech teaser.
- connect metrization theorems (urysohn metrization, nagata-smirnov pointers) to mit 18.901 problem set 3.

## reading cues

- munkres ch. 16–22.
- mit 18.901 lecture notes weeks 4–5; note any differences between ocw and current 2025 delivery.

## tasks

- [ ] table of spaces classified by separation level (sierpinski, co-countable, indiscrete, plane with line collapsed).
- [ ] proof sketches for urysohn and tietze with annotations referencing textbook theorems 18.1–18.3.
- [ ] metrization theorem crib notes with hypotheses vs conclusions.