Continuous maps between topological spaces

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

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Topological space — Both source and target have their own open families.
• Open and closed sets — Continuity checks target opens and their source preimages.
• 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.