Connected component of a topological space

Mathematical statement

Finite families enumerate every induced subspace and connected subset. The union of all connected subsets containing x is its unique inclusion-maximal component. Then check that the components are pairwise equal or disjoint, cover X and are closed in the stated finite model. Finitely many closed components also make each component open, a finite-specific conclusion. In a general space components need not be open: in the rational subspace of the real line they are singletons and are not open. Do not substitute globally largest cardinality, graph components or a different structure on the same carrier.

Representative formula (exact TeX)

K(X,x)=C

Assumptions

• Finite families supply the complete carrier and topology, not an adjacency graph standing in for open sets.
• A connected subset is tested in its induced topology for the absence of a nontrivial separation.

Symbol scope

• K — connected-component operator
• X — topological space
• x — basepoint
• C — inclusion-maximal connected subset

Intuition, definition, and misconceptions

Complete induced connectedness precedes inclusion-maximality, not maximum cardinality.

Audit the entire partition and closedness component by component.

Finite clopen components are a special case; the rational-subspace boundary protects the general nonopen distinction.

A largest-cardinality connected subset, graph component or finite-specific open component is not the general 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

• Connected topological space — Use connected-subspace semantics.
• Equivalence class — Retain the component partition.
• Subset — Specify inclusion-maximal connected subsets.
• Topological closure and interior — Retain the general closedness obligation.
• Connected component — Distinguish the same name on a different structure.
• Subspace topology — Use induced connectedness and the rational-subspace boundary.

Verification boundary

Finite enumeration certifies the specified topology; general maximality and closedness still need proof, and a nonopen finite-component witness cannot be fabricated.

not_formalized

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

Next obligation

Add quasicomponents and local 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.