Locally Closed Subset¶
A subset of a topological space that is open within its own closure, equivalently the intersection of an ambient-open and ambient-closed set.
Core Idea¶
A subset \(E\) of a topological space \(X\) is locally closed when it is open in its closure \(\overline E^X\). Equivalently, \(E=U\cap F\) for some ambient-open \(U\) and ambient-closed \(F\). This is one exact predicate with several equivalent tests, not a requirement that \(E\) itself be either open or closed in \(X\). The ambient topology must be stated.[^ref-bdd60a1677ee]
Scope of Application¶
In Euclidean \(\mathbb R\), \((0,1]\) is locally closed: it equals \((0,\infty)\cap(-\infty,1]\) and is open in its closure \([0,1]\), though neither open nor closed in \(\mathbb R\). In Zariski \(\mathbb A^2_{\mathbb C}\), the punctured \(x\)-axis \(V(y)\cap D(x)\) is similarly open within its closed support \(V(y)\). Conrad identifies the general algebraic form as closed \(Z\) intersect a polynomial nonvanishing open.[^ref-bdd60a1677ee]
Clarity¶
“Locally closed” means relatively open in the closure, or closed on a suitable neighborhood around each of its points; it does not mean globally open or globally closed. A finite union of locally closed pieces need not itself be one locally closed piece. In the Stacks Project, constructible subsets are finite unions of locally closed subsets in a Noetherian space; its general-space definition uses retrocompact-open qualifications.[ref-bdd60a1677ee][ref-71d6ee1ba547][^ref-db281184ae1b]
Manages Complexity¶
The closure-relative test replaces many pointwise checks with one set comparison; the \(U\cap F\) witness is convenient when equations and nonvanishing conditions already describe the set. The compression stops at the boundary of one piece: a union may need a separate witness for each part. A topological subset also does not, by itself, specify the sheaf or ideal data of a locally closed subscheme.[ref-bdd60a1677ee][ref-91c6ec041fc8][^ref-4f083c8ea6bb]
Abstract Reasoning¶
To classify \(E\), fix \(X\), compute \(\overline E^X\), and test whether \(E\) is open there—or exhibit ambient-open \(U\) and ambient-closed \(F\) with \(E=U\cap F\). If \(E\) is dense in \(X\) but not open in \(X\), it cannot be locally closed. If a later argument needs an arbitrary finite union or a scheme structure, state those stronger or different objects explicitly.[ref-bdd60a1677ee][ref-db281184ae1b][^ref-4f083c8ea6bb]
Knowledge Transfer¶
The same criterion applies literally in Euclidean and Zariski topology, though their open sets differ. It does not automatically transfer to a vague “open within support” metaphor outside topology. A portable skeleton is a future-prime question. The redirected Submaximal Space candidate is a distinct whole-space identity involving dense subsets and remains separately held.[ref-bdd60a1677ee][ref-0d351563c1a3]
[^ref-bdd60a1677ee]: Brian Conrad, “Dimension”, Stanford Math 145 handout, p. 1. [^ref-71d6ee1ba547]: The Stacks Project, “Constructible subsets”, Definition 5.15.1. [^ref-db281184ae1b]: The Stacks Project, “Constructible sets in Noetherian spaces”, Lemma 5.16.1. [^ref-91c6ec041fc8]: The Stacks Project, “Immersions of schemes”, §26.10. [^ref-4f083c8ea6bb]: The Stacks Project, “Reduced induced schemes”, §26.12. [^ref-0d351563c1a3]: Rostam Mohamadian, “Locally closed sets and submaximal spaces”, arXiv:2205.07191 (2022).
Neighborhood in Abstraction Space¶
Locally Closed Subset sits in a crowded region of the domain-specific corpus (38th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Point-Set Topology Foundations (17 abstractions)
Nearest neighbors
- Open Set — 0.89
- Clopen Set — 0.89
- Interior — 0.88
- Subspace Topology — 0.88
- A-paracompact Space — 0.88
Computed from structural-signature embeddings · 2026-10-08