branch of mathematics, a subset E {\displaystyle E} of a topological space X {\displaystyle X} is said to be locally closed if any of the following equivalent...

taken for locally compact, countably compact, metrizable, separable, countable; the second for locally connected. Locally closed subset A subset of a topological...

a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset. A...

not a complete subset). This shows that in a Hausdorff locally convex space that is not complete, the closed convex hull of compact subset might fail to...

closed immersion of schemes is a morphism of schemes f : Z → X {\displaystyle f:Z\to X} that identifies Z as a closed subset of X such that locally,...

examples show that a subset of a locally compact space need not be locally compact, which contrasts with the open and closed subsets in the previous section...

{\displaystyle A} is a clopen subset of Q . {\displaystyle \mathbb {Q} .} ( A {\displaystyle A} is not a clopen subset of the real line R {\displaystyle...

compact Hausdorff space is locally compact, a connected space—and even a connected subset of the Euclidean plane—need not be locally connected (see below)...

Cartesian closed, then it is called locally cartesian closed. Note that if C is locally Cartesian closed, it need not actually be Cartesian closed; that happens...

{\displaystyle \Gamma _{x}\subset \Gamma '_{x}} where the equality holds if X {\displaystyle X} is compact Hausdorff or locally connected. A space in which...

{\displaystyle f:X\to Y} maps closed subsets of its domain to closed subsets of its codomain; that is, for any closed subset C {\displaystyle C} of X , {\displaystyle...

locally closed subset). It states: let V be a submanifold of a manifold M and U a tubular neighborhood of V. If σ {\displaystyle \sigma } is a closed...

such that X is the union of a finite number of locally closed subsets on each of which the sheaf is a locally constant sheaf. It has its origins in algebraic...

} i.e. its Lebesgue integral is finite on all compact subsets K of Ω, then f is called locally integrable. The set of all such functions is denoted by...

simultaneously be both an open subset and a closed subset. Such subsets are known as clopen sets. Explicitly, a subset S {\displaystyle S} of a topological...

well-known fact that a subset S ⊆ X {\displaystyle S\subseteq X} is closed in X {\displaystyle X} if and only if it is "locally closed in X {\displaystyle...

{G}}_{x}} otherwise. More generally, if A ⊂ X {\displaystyle A\subset X} is a locally closed subset, then there exists an open U {\displaystyle U} of X {\displaystyle...

compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures"...

_{t>0}t(S-s_{0}).} Theorem (Dieudonné). Let A and B be non-empty, closed, and convex subsets of a locally convex topological vector space such that rec ⁡ A ∩ rec...

introduced by Bourbaki in 1948. The empty set is always a closed nowhere dense (and thus meagre) subset of every topological space. In the nonmeagre space X...

{\displaystyle V} are open and retrocompact subsets of X {\displaystyle X} . A subset Z ⊂ X {\displaystyle Z\subset X} is locally constructible if there is a cover...

containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane...

Equivalently, if X is a locally compact metric space, then f is locally Lipschitz if and only if it is Lipschitz continuous on every compact subset of X. In spaces...

of rational numbers Q is not locally compact if given the relative topology as a subset of the real numbers. It is locally compact if given the discrete...

A collection of subsets of a topological space X {\displaystyle X} is said to be locally finite if each point in the space has a neighbourhood that intersects...

Each stratum X {\displaystyle X} is a locally closed subset and the decomposition S {\displaystyle S} is locally finite. The decomposition S {\displaystyle...

by M + 2 {\displaystyle M+2} . Lemma: A closed subset of a compact set is compact. Let K be a closed subset of a compact set T in Rn and let CK be an...

is closed in X . {\displaystyle X.} The sum of a compact set and a closed set is closed. However, the sum of two closed subsets may fail to be closed (see...

mathematics, a topological space X is called a regular space if every closed subset C of X and a point p not contained in C have non-overlapping open neighborhoods...

irrational. Then H is dense in G and hence not closed. In the relative topology, a small open subset of H is composed of infinitely many almost parallel...