---
title: "Topology"
description: "7 topics in Mathematics, in the order they build on each other."
canonical: https://lightmysky.com/learn/mathematics/areas/topology
source: https://lightmysky.com/learn/mathematics/areas/topology.md
retrieved: 2026-09-02
---

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# Topology

7 topics in Mathematics, in the order they build on each other.

Page: https://lightmysky.com/learn/mathematics/areas/topology

- [Metric Spaces: Distance as an Axiom](https://lightmysky.com/learn/mathematics/metric-spaces-distance-as-an-axiom-mt_A3TMb8wL6k): Keep only the properties distance must have, and check how much of the analysis of the real line survives on that alone.
- [Open Sets, Closed Sets and Limit Points](https://lightmysky.com/learn/mathematics/open-sets-closed-sets-and-limit-points-mt_9bBdT38dbB): Rebuild openness, closure and limit points from balls alone, and restate continuity as preimages of open sets.
- [Completeness and the Contraction Mapping Theorem](https://lightmysky.com/learn/mathematics/completeness-and-the-contraction-mapping-theorem-mt_DYsLzg-IaK): Say what a complete metric space is, and prove that a contraction on one has exactly one fixed point that iteration finds.
- [Compactness in Metric Spaces](https://lightmysky.com/learn/mathematics/compactness-in-metric-spaces-mt_dBLXU0wSN9): Compare the open-cover and sequential definitions, prove they agree in a metric space, and see what compactness buys for continuous functions.
- [Topological Spaces: Continuity Without Distance](https://lightmysky.com/learn/mathematics/topological-spaces-continuity-without-distance-mt_D2oPq0AVhI): Declare which sets count as open and continuity can be defined with no distance anywhere in sight. One set can carry several topologies, and the choice decides which functions are continuous.
- [Compactness and Connectedness](https://lightmysky.com/learn/mathematics/compactness-and-connectedness-mt_ybV1lUedHs): Two properties that continuous maps preserve, which is why proofs lean on them so heavily. Compactness turns any open cover into a finite one, and connectedness rules out splitting a space in two.
- [The Fundamental Group and Loops on a Circle](https://lightmysky.com/learn/mathematics/the-fundamental-group-and-loops-on-a-circle-mt_zH50_L0nu4): Loops based at a point, counted up to continuous deformation, form a group. Computing it for the circle is what proves a disc and an annulus are genuinely different spaces rather than merely drawn differently.