Open and closed sets

Mathematical statement

In a topological space (X,τ), a subset U⊆X is open exactly when U∈τ, and C⊆X is closed exactly when X\C∈τ. Open and closed are not mutually exclusive; ∅ and X are always both open and closed.

Representative formula (exact TeX)

C\setminus U

Assumptions

• The underlying set X and its topology τ are fixed.
• Every complement is relative to the same X; first qualify each candidate as a subset of X.

Symbol scope

• C — candidate closed set
• X — underlying set
• U — candidate open set

Intuition, definition, and misconceptions

Closed does not mean not open. A set may be open only, closed only, both, or neither.

Arbitrary unions and finite intersections of open sets are open. Dually, arbitrary intersections and finite unions of closed sets are closed. A finite model permits complete open, closed and clopen inventories.

Complement duality requires one fixed underlying set. Changing the universe silently changes the topological context of the classification.

Do not interpret closed as not open or omit the underlying set relative to which complements are taken.

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 fixed topology supplies the open family.
• Subset — Open and closed candidates are subsets of the underlying set.
• Set operations — The closed-set definition uses relative complements.

Verification boundary

Finite topologies permit complete enumeration; this does not replace general formal proofs of arbitrary intersection/union closure.

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 IsOpen, IsClosed, complement and the finite/arbitrary closed-set closure theorems.

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.