# Topological Space

**Type:** Concept  
**Domain:** Topology  
**Codex URL:** /mathematics/topology/topological-space/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
A set with a collection of open sets satisfying 3 axioms, giving a notion of nearness without requiring distance. Foundation of topology.

## Definition
(X,τ) where τ (topology) satisfies: ∅,X∈τ; arbitrary unions of τ-sets ∈τ; finite intersections of τ-sets ∈τ.

## Continuity redefined
f:X→Y continuous iff preimage of every open set is open — no distances needed.

## Homeomorphism
Continuous bijection with continuous inverse — "topologically the same" (doughnut = coffee cup).

## Historical development
Euler (1736, 1750s polyhedron formula) → Poincaré "analysis situs" (1895) → Hausdorff axiomatic definition (1914) → Kuratowski closure-operator equivalent (1922).

## Sources
### Tier 1
- Munkres, J.R. (2000). *Topology*. 2nd ed.
- Hausdorff, F. (1914). *Grundzüge der Mengenlehre*.
### Tier 2
- Hatcher, A. (2002). *Algebraic Topology*.
### Tier 3
- James, I.M. (ed.) (1999). *History of Topology*.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*