---
date: '2025-11-01'
description: evergreen hub for topology study anchored on munkres and mit 18.901.
id: topology
modified: 2026-05-24 12:53:26 GMT-04:00
seealso:
  - '[[thoughts/topology/point set|point-set topology]]'
  - '[[thoughts/topology/separation|separation]]'
  - '[[thoughts/topology/compactness|compactness]]'
  - '[[thoughts/topology/fundamental group|fundamental group]]'
  - '[[thoughts/topology/algebraic bridge|algebraic bridge]]'
  - '[[thoughts/topology/simply connected|simple connectivity]]'
  - '[[thoughts/topology/differential foundations|differential foundations]]'
  - '[[thoughts/topology/3 manifolds|3-manifold topology]]'
  - '[[thoughts/topology/ricci flow|ricci flow]]'
  - '[[thoughts/topology/resources|resources]]'
socials:
  ocw: https://ocw.mit.edu/courses/18-901-introduction-to-topology-fall-2004/
tags:
  - math
  - math/topology
  - evergreen
title: topology
created: '2025-11-01'
published: '2025-11-01'
pageLayout: default
slug: thoughts/topology
permalink: https://aarnphm.xyz/thoughts/topology.md
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---
I’m using {{sidenotes[munkres' topology]: james munkres, [[thoughts/pdfs/munkres-topology.pdf|Topology]] (2nd ed.)}} and Amstrong’s [[library/Basic Topology]]

- foundation:
  - set theory, logic, proof skills, and metric intuition
  - see [[thoughts/norm|norm]] for metric-induced topology
- **phase 1 — point-set core** (weeks 2–5): topological spaces, bases, subspaces, product and quotient topologies (munkres ch. 2–5). track with [[courses/18.901-fall-2004/|mit 18.901]] weeks 1–3.
- **phase 2 — separation + countability** (weeks 6–7): separation axioms, urysohn lemma, metrization theorems (munkres ch. 16–22). mirror mit 18.901 weeks 4–5.
- **phase 3 — compactness + connectedness** (weeks 8–9): compactness, lindelöf, local compactness, connectedness (munkres ch. 26–31). align with mit 18.901 weeks 6–7 problem sets.
- **phase 4 — algebraic entrance** (weeks 10–12): fundamental group, covering spaces, van kampen (munkres ch. 51–56). follow mit 18.901 weeks 8–11 lectures.
- **phase 5 — beyond** (weeks 13+): transition into homology and homotopy (munkres appendix, hatcher ch. 0–1) before stepping into mit 18.906.

## poincaré conjecture

I want to understand [[thoughts/topology/poincare|poincaré conjecture]].

[[thoughts/topology/simply connected|simple connectivity]] ($\pi_1=0$) is strictly stronger than $H_1=0$ (see poincaré homology sphere). this makes it the “right” topological condition.