# Continuous maps between topological spaces

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A map is topologically continuous when every open set in its target has an open preimage in its source.

mko-topological-continuity · definition · v0.10.0



## Mathematical statement

For topological spaces (X,τX), (Y,τY) and a total map f:X→Y, f is continuous exactly when f⁻¹[V]∈τX for every V∈τY. This is a preimage condition; it does not require images of open sets to be open.





## Representative formula (exact TeX)

f:X\to Y





## Assumptions


- The source and target underlying sets and both topologies are fixed.
- The map f is supplied by a complete total table; the preimage collects all source elements mapping into the target subset.





## Symbol scope


- f — total map between topological spaces
- X — source space
- Y — target space





## Intuition, definition, and misconceptions

For finite topologies, enumerate every target open and compute its complete preimage to decide continuity of the given table.

Compositions of continuous maps are continuous. A finite model can recompute both maps and every open-set preimage; sampling individual points is insufficient.

An open map sends open sets to open sets. This differs from continuity: a map may be continuous but not open, or open but not continuous.

Do not define continuity by open images or check only one target open.





## Adopted derivation guide

[Continuous maps between topological spaces: deciding continuity by preimages in finite spaces and separating it from openness](https://ocme.evemisslab.com/en/derivations/derivation-mko-topological-continuity-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Topological space](https://ocme.evemisslab.com/en/mko/mko-topological-space/) — Both source and target have their own open families.
- [Open and closed sets](https://ocme.evemisslab.com/en/mko/mko-open-closed-set/) — Continuity checks target opens and their source preimages.
- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — The full total map defines the preimage.





## Verification boundary

Complete finite topologies support exact decisions; no automatic equivalence with every semantic detail of the existing real-function continuity entry is claimed.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Bind the open-preimage definition of Mathlib Continuous and separately bridge real functions, subspaces and isolated points.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/topological-continuity.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-topological-continuity/index.md)