Continuous image of a compact set

Mathematical statement

Pull an ambient-open cover of f(K) back along f. Continuity makes preimages open and they cover K. A finite subcover of K pushes forward using the same cover indices to cover f(K). Injectivity and surjectivity are not required. Compactness of an image does not imply compactness of the domain. Compact preimages and closed images require additional hypotheses, often Hausdorff for closedness.

Representative formula (exact TeX)

C(f(K))

Assumptions

• Positive companions declare complete finite source/target topologies, a total map and complete open covers.
• K uses subspace compactness, and cover members belong to the target topology.

Symbol scope

• C — compactness predicate
• f — continuous total map
• K — compact subset

Intuition, definition, and misconceptions

Pull the image cover back to a relative-open cover of K.

Preserve the finite-subcover indices when returning to the target cover.

The theorem has a forward implication, not an unconditional converse.

A compact image does not force a compact domain, and an image is not automatically closed without appropriate extra hypotheses.

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

• Compact subset — Source compactness is required.
• Continuous maps between topological spaces — Open preimages express continuity.
• Function mapping — Images and preimages use a total map.

Verification boundary

Finite positive certificates do not enumerate all infinite topologies. A separate constant map from a standard real open interval provides the converse counterexample.

not_formalized

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

Next obligation

Add Hausdorff closedness and quotient topology.

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.