 | This article does not cite any sources .(December 2020) (Learn how and when to remove this template message) |
In topology, the exterior of a subset of a topological space is the union of all open sets of which are disjoint from It is itself an open set and is disjoint from The exterior of in is often denoted by or, if is clear from context, then possibly also by or
The exterior is equal to the complement of the (topological) closure of and to the (topological) interior of the complement of in
The topological exterior of a subset always satisfies:
and as a consequence, many properties of can be readily deduced directly from those of the interior and elementary set identities. Such properties include the following:
Unlike the interior operator, is not idempotent, although it does have the following property:
 | This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it. |
In mathematics, a continuous function is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to sufficiently small changes in its input. If not continuous, a function is said to be discontinuous. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, during which attempts such as the epsilon–delta definition were made to formalize it.
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; in that sense the product topology is the natural topology on the Cartesian product.
This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.
In mathematics, open sets are a generalization of open intervals in the real line. In a metric space—that is, when a distance is defined—open sets are the sets that, with every point P, contain all points that are sufficiently near to P.
In mathematics, the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S. The closure of S may equivalently be defined as the union of S and its boundary, and also as the intersection of all closed sets containing S. Intuitively, the closure can be thought of as all the points that are either in S or "near" S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior.
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation. This should not be confused with a closed manifold.
In mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered anywhere. For example, the integers are nowhere dense among the reals, whereas an open ball is not.
In mathematics, specifically in topology, the interior of a subset S of a topological space X is the union of all subsets of S that are open in X. A point that is in the interior of S is an interior point of S.
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it 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 bd(S), fr(S), 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 Baire space is a topological space such that every intersection of a countable collection of open dense sets in the space is also dense. Complete metric spaces and locally compact Hausdorff spaces are examples of Baire spaces according to the Baire category theorem. The spaces are named in honor of René-Louis Baire who introduced the concept.
In mathematics, a limit point of a set in a topological space is a point that can be "approximated" by points of in the sense that every neighbourhood of with respect to the topology on also contains a point of other than itself. A limit point of a set does not itself have to be an element of
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function is open if for any open set in the image is open in Likewise, a closed map is a function that maps closed sets to closed sets. A map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa.
A semiregular space is a topological space whose regular open sets form a base.
In mathematics, a topological space is usually defined in terms of open sets. However, there are many equivalent characterizations of the category of topological spaces. Each of these definitions provides a new way of thinking about topological concepts, and many of these have led to further lines of inquiry and generalisation.
In topology and related fields of mathematics, a sequential space is a topological space that satisfies a very weak axiom of countability.
In topology and other branches of mathematics, a topological space X is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.
In the mathematical field of topology, a hyperconnected space or irreducible space is a topological space X that cannot be written as the union of two proper closed sets. The name irreducible space is preferred in algebraic geometry.
In the field of topology, a Fréchet–Urysohn space is a topological space with the property that for every subset the closure of in is identical to the sequential closure of in Fréchet–Urysohn spaces are a special type of sequential space.
In topology, a subfield of mathematics, filters are special families of subsets of a set that can be used to study topological spaces and define all basic topological notions such a convergence, continuity, compactness, and more. Filters also provide a common framework for defining various types of limits of functions such as limits from the left/right, to infinity, to a point or a set, and many others. Special types of filters called ultrafilters have many useful technical properties and they may often be used in place of arbitrary filters.
