---
date: '2025-11-01'
description: compactness, connectedness, lindelöf, local compactness aligned with mit 18.901 weeks 6–7.
id: topology-compactness
modified: 2026-06-05 15:08:22 GMT-04:00
seealso:
  - '[[thoughts/topology/separation|separation axioms]]'
  - '[[thoughts/topology/fundamental group|fundamental group]]'
tags:
  - math
  - math/topology
title: compactness and connectedness
created: '2025-11-01'
published: '2025-11-01'
pageLayout: default
slug: thoughts/topology/compactness
permalink: https://aarnphm.xyz/thoughts/topology/compactness.md
generator:
  quartz: v4.6.0
  hostedProvider: Cloudflare
  baseUrl: aarnphm.xyz
full: https://aarnphm.xyz/llms-full.txt
---
## goals

- distill definitions: compact, sequentially compact, countably compact, lindelöf, locally compact.
- summarize connectedness, path-connectedness, components, local connectedness.
- catalogue mit 18.901 applications (heine-borel, tikhonov sketches, bounded continuous functions).

## reading cues

- munkres ch. 26–31.
- mit 18.901 lectures 12–15 (pset 5 focus).

## tasks

- [ ] prove equivalences between compactness notions in metric spaces (munkres 27.3).
- [ ] work through alexandroff one-point compactification examples.
- [ ] record connectedness counterexamples (topologist’s sine curve, comb space).