---
title: "Topological Spaces: Continuity Without Distance"
description: "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."
canonical: https://lightmysky.com/learn/mathematics/topological-spaces-continuity-without-distance-mt_D2oPq0AVhI
source: https://lightmysky.com/learn/mathematics/topological-spaces-continuity-without-distance-mt_D2oPq0AVhI.md
retrieved: 2026-09-12
---

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# Topological Spaces: Continuity Without Distance

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.

Subject: Mathematics · Area: Topology · Ages 23 to 24
Page: https://lightmysky.com/learn/mathematics/topological-spaces-continuity-without-distance-mt_D2oPq0AVhI

## Ready when they can

- Check the axioms for a proposed collection of open sets
- Define continuity by preimages and reconcile it with the epsilon-delta version
- Give two topologies on one set and a function continuous for one but not the other

## Lesson: Continuity with open sets

A topology declares which sets count as open. It must hold the empty set and the whole set, stay closed under arbitrary unions, and stay closed under finite intersections. On the set with points 1 and 2, the collection holding the empty set, the set with 1 alone, and the whole set passes all three tests.

Continuity then needs no ruler: a function is continuous when the preimage of every open set is open. Images go the wrong way and need not cooperate, so never test continuity with images of opens. Unions are unrestricted while intersections stay finite, which is the asymmetry to memorize.

**Example.** On the real line this new wording agrees with epsilon-delta continuity. Every open set there is built from open intervals, so controlling outputs by restricting inputs becomes a preimage condition. Learn both dialects so you recognize the same idea far from any metric.

**Tip.** One set can carry several topologies, and the choice decides which functions are continuous. The identity map from a set with the discrete topology to the same set with the indiscrete topology is continuous, while the reverse map is not. Restricting a domain works the same way: deleting bad points can create continuity.

**Recap.** A topology is three axioms on open sets, continuity means open preimages, and the same function can change status when the topology changes.

## Practice

18 questions on this page, each with its working shown.

## Needs first

- [The Galois Correspondence](https://lightmysky.com/learn/mathematics/the-galois-correspondence-mt_30kyliQR_L)
- [Metric Spaces: Distance as an Axiom](https://lightmysky.com/learn/mathematics/metric-spaces-distance-as-an-axiom-mt_A3TMb8wL6k)
- [Compactness in Metric Spaces](https://lightmysky.com/learn/mathematics/compactness-in-metric-spaces-mt_dBLXU0wSN9)
- [Continuity and Uniform Continuity](https://lightmysky.com/learn/mathematics/continuity-and-uniform-continuity-mt_plWc9raOwz)

## Opens up

- [Tensors and the Metric on a Curved Manifold](https://lightmysky.com/learn/science/tensors-and-the-metric-on-a-curved-manifold-mt_9r9SG3NcZR)
- [Definitions as Design Decisions](https://lightmysky.com/learn/mathematics/definitions-as-design-decisions-mt_b6ip8eP0-d)
- [Compactness and Connectedness](https://lightmysky.com/learn/mathematics/compactness-and-connectedness-mt_ybV1lUedHs)