# Subspace topology

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

For Y⊆X, the relative topology consists of every intersection of an ambient open set with Y.

mko-subspace-topology · definition · v0.10.0



## 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

[Relative openness differs from ambient openness](https://ocme.evemisslab.com/en/derivations/derivation-mko-subspace-topology-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Topological space](https://ocme.evemisslab.com/en/mko/mko-topological-space/) — The parent topology is complete.
- [Subset](https://ocme.evemisslab.com/en/mko/mko-subset/) — The inclusion Y⊆X is fixed.
- [Topological basis](https://ocme.evemisslab.com/en/mko/mko-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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/subspace-topology.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-subspace-topology/index.md)