In mathematics, specifically general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples of compact spaces include a closed real interval, a union of a finite number of closed intervals, a rectangle, or a finite set of points. This notion is defined for more general topological spaces in various ways, which are usually equivalent in Euclidean space but may be inequivalent in other spaces.
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces.
In mathematics, in general topology, compactification is the process or result of making a topological space into a compact space. A compact space is a space in which every open cover of the space contains a finite subcover. The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape".
In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.
In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. These conditions are examples of separation axioms and their further strengthenings define completely normal Hausdorff spaces, or T5 spaces, and perfectly normal Hausdorff spaces, or T6 spaces.
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set R of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one. It can be thought of as a vector space, a metric space, a topological space, a measure space, or a linear continuum.
In mathematics, a curve is an object similar to a line, but that does not have to be straight.
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has a neighborhood not containing the other point. An R0 space is one in which this holds for every pair of topologically distinguishable points. The properties T1 and R0 are examples of separation axioms.
In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.
In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover.
In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square. Because Giuseppe Peano (1858–1932) was the first to discover one, space-filling curves in the 2-dimensional plane are sometimes called Peano curves, but that phrase also refers to the Peano curve, the specific example of a space-filling curve found by Peano.
In mathematics, a cofinite subset of a set is a subset whose complement in is a finite set. In other words, contains all but finitely many elements of If the complement is not finite, but it is countable, then one says the set is cocountable.
In mathematics, a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets. These conditions guarantee that the measure is "compatible" with the topology of the space, and most measures used in mathematical analysis and in number theory are indeed Radon measures.
In topology, the Sorgenfrey plane is a frequently-cited counterexample to many otherwise plausible-sounding conjectures. It consists of the product of two copies of the Sorgenfrey line, which is the real line under the half-open interval topology. The Sorgenfrey line and plane are named for the American mathematician Robert Sorgenfrey.
Counterexamples in Topology is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr.
In mathematics, a space is a set with some added structure.
In mathematics, particularly topology, a Gδ space is a topological space in which closed sets are in a way ‘separated’ from their complements using only countably many open sets. A Gδ space may thus be regarded as a space satisfying a different kind of separation axiom. In fact normal Gδ spaces are referred to as perfectly normal spaces, and satisfy the strongest of separation axioms.