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...

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...

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...

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...

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)...

{\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...

{\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...

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...

{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...

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

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,...

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...

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...

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"...

vector space (TVS) is a subset that is a closed absorbing disk; that is, a barrel is a convex, balanced, closed, and absorbing subset. Every barrel must contain...

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...

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...

two norm-closed *-ideals: I0 = {0} and the ideal K = K(H) of compact operators. Thus as a set, Prim(L(H)) = {I0, K}. Now {K} is a closed subset of Prim(L(H))...

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...

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...

{\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...

essentially equivalent to the question of whether an orbit space of some locally closed subset of the Hilbert or Chow schemes by the projective group exists. To...

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...

vanishing on that closed subset. ind-scheme An ind-scheme is an inductive limit of closed immersions of schemes. invertible sheaf A locally free sheaf of...

{\displaystyle M+2} . Lemma: A closed subset of a compact set is compact. Let K {\displaystyle K} be a closed subset of a compact set T {\displaystyle...

non-compact metric spaces is never normal. Every closed subset of a normal space is normal. The continuous and closed image of a normal space is normal. The main...

mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean n-spaces. For lower dimensions n=1, 2,...

topological space ( X , T ) {\displaystyle (X,{\mathcal {T}})} and closed subset S ⊆ X , {\displaystyle S\subseteq X,} the subspace ( S , T | S ) {\displaystyle...

is contained in some open (or closed) ball centered at the origin (zero). Any translation, scalar multiple, and subset of a bounded set is again bounded...