Connected topological space

Mathematical statement

In a complete finite topology, enumerate all candidate separations or clopen subsets. Disconnectedness requires an explicit witness and connectedness requires excluding every one. The empty space is connected under the declared vacuous convention, and every singleton space is connected. Do not conflate this with graph connectivity.

Representative formula (exact TeX)

C(X)

Assumptions

• The finite carrier and topology are complete.
• The empty-space convention is explicitly declared as connected.

Symbol scope

• C — connected-space record
• X — finite topological space

Intuition, definition, and misconceptions

A separation needs two nonempty disjoint open sets whose union is the whole carrier.

A nontrivial clopen subset supplies an equivalent disconnection witness.

Connectedness requires exhaustive exclusion; empty and singleton boundaries have explicit conventions.

A few sampled opens do not certify connectedness, and graph connectivity is not the same definition.

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 — The complete open-set family is explicit.
• Open and closed sets — Connectedness has the clopen criterion.
• Set operations — Separation uses disjointness, union and complement.

Verification boundary

Complete finite enumeration certifies only the specified topology.

not_formalized

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

Next obligation

Add components and path connectedness.

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.