# Topological space

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

A topological space consists of an underlying set and a family of open sets containing the empty set and whole space and closed under arbitrary unions and finite intersections.

mko-topological-space · definition · v0.10.0



## Mathematical statement

For a set X, a topology on X is a family τ⊆P(X) such that ∅∈τ, X∈τ, the union of any subfamily of τ belongs to τ, and the intersection of any finite subfamily of τ belongs to τ. The pair (X,τ) is a topological space.





## Representative formula (exact TeX)

T\subseteq P





## Assumptions


- The underlying set X is fixed, and every member of τ is a subset of X.
- Arbitrary unions include the empty family, and finite intersections include the empty family; interpret the empty intersection as X.





## Symbol scope


- T — represents the family τ of open sets
- P — represents the powerset P(X) of the underlying set X





## Intuition, definition, and misconceptions

A topology specifies which subsets are called open; it does not first specify a distance. The discrete topology declares every subset open, while the indiscrete topology contains only the empty set and the whole space.

For a finite underlying set, one can enumerate the complete candidate family and check all closure requirements. Testing only a few unions or intersections does not establish that the candidate is a topology.

The same set can carry different topologies. If τ₁⊆τ₂, then τ₂ is finer; this compares families of open sets, not the sizes of the underlying sets.

Mistakes include treating any family of subsets as a topology or confusing closure under binary unions alone with the general arbitrary-union requirement.





## Adopted derivation guide

[Topological space: fully verifying a chain of open sets on a three-point set, and recognising non-topologies](https://ocme.evemisslab.com/en/derivations/derivation-mko-topological-space-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Set](https://ocme.evemisslab.com/en/mko/mko-set/) — The underlying set and family of open sets both use set semantics.
- [Set operations](https://ocme.evemisslab.com/en/mko/mko-set-operations/) — Topology axioms use unions, finite intersections, the empty set and the whole space.





## Verification boundary

Complete powerset enumeration permits exact checks on a finite underlying set; it does not replace a general formal proof of the arbitrary-union axiom.

not_formalized

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





## Next obligation

Bind Mathlib TopologicalSpace and its IsOpen, univ, binary-intersection and sUnion axioms, and construct an explicit semantic bridge to finite enumeration.





## 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/topological-space.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-topological-space/index.md)