Neighborhood

Mathematical statement

In a topological space (X,τ), N⊆X is a neighborhood of x∈X exactly when there is U∈τ with x∈U⊆N. The open sets containing x form an open neighborhood base, but a general neighborhood need not itself be open.

Representative formula (exact TeX)

x\in N

Assumptions

• X, τ, x and candidate N are explicit, with x∈X and N⊆X.
• Neighborhoods use containment of an open set; no distance, radius or single least neighborhood is assumed.

Symbol scope

• x — specified point
• N — candidate neighborhood

Intuition, definition, and misconceptions

To test N, search all open witnesses containing x and contained in N. The candidate itself can be nonopen and still be a neighborhood.

Neighborhoods are upward closed: if N is a neighborhood of x and N⊆M⊆X, then M is also a neighborhood of x.

A finite topology permits a complete inventory of a point’s neighborhoods. This inventory depends on both topology and point, not just the underlying set.

Do not identify neighborhoods with open sets or open balls, or infer the neighborhood condition from x∈N alone.

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 — Neighborhoods are derived from the open family.
• Open and closed sets — The witness U must be open.
• Set membership — The point must belong to the open witness and candidate neighborhood.

Verification boundary

Complete finite powerset enumeration checks the specified model; no metric-ball or general filter theorem is inferred.

not_formalized

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

Next obligation

Bind nhds, nhds_basis_opens and the upward-closure semantics of neighborhood filters.

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.