In topology, a branch of mathematics, a subset of a topological space is said to be locally closed if any of the following equivalent conditions are satisfied: [1] [2] [3] [4]
The second condition justifies the terminology locally closed and is Bourbaki's definition of locally closed. [1] To see the second condition implies the third, use the facts that for subsets is closed in if and only if and that for a subset and an open subset
The interval is a locally closed subset of For another example, consider the relative interior of a closed disk in It is locally closed since it is an intersection of the closed disk and an open ball.
On the other hand, is not a locally closed subset of .
Recall that, by definition, a submanifold of an -manifold is a subset such that for each point in there is a chart around it such that Hence, a submanifold is locally closed. [5]
Here is an example in algebraic geometry. Let U be an open affine chart on a projective variety X (in the Zariski topology). Then each closed subvariety Y of U is locally closed in X; namely, where denotes the closure of Y in X. (See also quasi-projective variety and quasi-affine variety.)
Finite intersections and the pre-image under a continuous map of locally closed sets are locally closed. [1] On the other hand, a union and a complement of locally closed subsets need not be locally closed. [6] (This motivates the notion of a constructible set.)
Especially in stratification theory, for a locally closed subset the complement is called the boundary of (not to be confused with topological boundary). [2] If is a closed submanifold-with-boundary of a manifold then the relative interior (that is, interior as a manifold) of is locally closed in and the boundary of it as a manifold is the same as the boundary of it as a locally closed subset. [2]
A topological space is said to be submaximal if every subset is locally closed. See Glossary of topology#S for more of this notion.
In mathematics, any vector space has a corresponding dual vector space consisting of all linear forms on together with the vector space structure of pointwise addition and scalar multiplication by constants.
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In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set S include and . Some authors use the term frontier instead of boundary in an attempt to avoid confusion with a different definition used in algebraic topology and the theory of manifolds. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, Metric Spaces by E. T. Copson uses the term boundary to refer to Hausdorff's border, which is defined as the intersection of a set with its boundary. Hausdorff also introduced the term residue, which is defined as the intersection of a set with the closure of the border of its complement.
In mathematics, a diffeology on a set generalizes the concept of smooth charts in a differentiable manifold, declaring what the "smooth parametrizations" in the set are.
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In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.
In mathematics, a submanifold of a manifold is a subset which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required. Different authors often have different definitions.
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In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real n-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold. Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure. However, not every topological manifold can be endowed with a particular additional structure. For example, the E8 manifold is a topological manifold which cannot be endowed with a differentiable structure.
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In topology and related areas of mathematics, a subset A of a topological space X is said to be dense in X if every point of X either belongs to A or else is arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it. Formally, is dense in if the smallest closed subset of containing is itself.