Topological space

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

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Set — The underlying set and family of open sets both use set semantics.
• 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.