---
date: '2025-11-01'
description: scaffolding (munkres ch. 2–5).
id: point set
modified: 2026-06-05 15:08:21 GMT-04:00
seealso:
  - '[[thoughts/topology|topology hub]]'
  - '[[thoughts/topology/separation|separation axioms]]'
tags:
  - math
  - math/topology
title: point-set topology
created: '2025-11-01'
published: '2025-11-01'
pageLayout: default
slug: thoughts/topology/point-set
permalink: https://aarnphm.xyz/thoughts/topology/point-set.md
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---
## objectives

- internalize definitions of topological spaces, bases, subbases.
- compute product, subspace, and quotient topologies through textbook exercises.
- sync with mit 18.901 lectures 1–9; annotate key lemmas and worked examples.

## reading cues

- munkres ch. 2: topological spaces and basis constructions.
- munkres ch. 3: subspace, product, quotient, metric-generated topologies.
- munkres ch. 4–5: continuity, homeomorphisms, metric/first countability.

## mit 18.901 alignment

- lecture 1–3: definitions + examples; capture boardwork snapshots and translate into examples here.
- lecture 4–6: product topologies, projection maps, universal properties.
- lecture 7–9: quotient topology, identification spaces, classic counterexamples.

## checklist

- [ ] build example catalogue: standard, discrete, indiscrete, finite complement, lower limit line.
- [ ] prove basis-subspace relationships for selected exercises (e.g., munkres 3.2, 3.3).
- [ ] implement product topology computations for torus and mobius strip.
- [ ] summarize quotient topology pitfalls using mit pset 2.