Subspace topology

Mathematical statement

The induced topology is {U∩Y:U∈tau_X}, normalizing duplicate and empty intersections. Relative opens need not be ambient open and their witnesses U need not lie inside Y. Relative closedness uses open complements inside Y, or intersections with ambient closed sets. Y is the relative universe in every topology obligation.

Representative formula (exact TeX)

T(Y)

Assumptions

• The complete finite ambient carrier, complete parent topology and subset Y are explicit.
• Every candidate relative set is a subset of Y.

Symbol scope

• T — subspace topology
• Y — subspace carrier

Intuition, definition, and misconceptions

Every relative open is O∩Y for an ambient open O.

Complements are taken inside Y.

Relative and ambient openness must not be conflated.

Do not require the ambient open O to lie in Y or declare every relative open open in X.

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 parent topology is complete.
• Subset — The inclusion Y⊆X is fixed.
• Topological basis — Ambient basis elements can be intersected with Y.

Verification boundary

Complete enumeration certifies only the declared finite 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 compact subsets and product 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.