Construction
The open sets in this topology are taken to be the whole set S, the empty set ∅, and the sets generated by
The sets generated by Xn will be formed by all possible unions of finite intersections of the Xn. [3]
In mathematics, and especially general topology, the interlocking interval topology is an example of a topology on the set S := R+ \ Z+, i.e. the set of all positive real numbers that are not positive whole numbers. [1] To give the set S a topology means to say which subsets of S are "open", and to do so in a way that the following axioms are met: [2]
The open sets in this topology are taken to be the whole set S, the empty set ∅, and the sets generated by
The sets generated by Xn will be formed by all possible unions of finite intersections of the Xn. [3]
In mathematics, more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space in various ways.
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M.
In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.
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 topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.
The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space.
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.
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 the mathematical field of topology, a Gδ set is a subset of a topological space that is a countable intersection of open sets. The notation originated in Germany with G for Gebiet meaning open set in this case and δ for Durchschnitt. The term inner limiting set is also used. Gδ sets, and their dual, Fσ sets, are the second level of the Borel hierarchy.
In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set of real numbers; it is different from the standard topology on and has a number of interesting properties. It is the topology generated by the basis of all half-open intervals [a,b), where a and b are real numbers.
In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the set is cocountable.
In mathematics, a hyperconnected 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 mathematics a topological space is called countably compact if every countable open cover has a finite subcover.
In mathematics, a topological space X is said to be limit point compact or weakly countably compact if every infinite subset of X has a limit point in X. This property generalizes a property of compact spaces. In a metric space, limit point compactness, compactness, and sequential compactness are all equivalent. For general topological spaces, however, these three notions of compactness are not equivalent.
In topology and related areas of mathematics, a subset A of a topological space X is called dense if every point x in X either belongs to A or is a limit point of A; that is, the closure of A is constituting the whole set X. Informally, for every point in X, the point is either in A or 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.
In mathematics, and especially general topology, the prime integer topology and the relatively prime integer topology are examples of topologies on the set of positive whole numbers, i.e. the set Z+ = {1, 2, 3, 4, …}. To give the set Z+ a topology means to say which subsets of Z+ are "open", and to do so in a way that the following axioms are met:
In mathematics, more specifically general topology, the nested interval topology is an example of a topology given to the open interval (0,1), i.e. the set of all real numbers x such that 0 < x < 1. The open interval (0,1) is the set of all real numbers between 0 and 1; but not including either 0 or 1.
In mathematics, more specifically general topology, the divisor topology is an example of a topology given to the set of positive integers that are greater than or equal to two, i.e., . The divisor topology is the poset topology for the partial order relation of divisibility on the positive integers.
In general topology, a branch of mathematics, the evenly spaced integer topology is the topology on the set of integers = {…, −2, −1, 0, 1, 2, …} generated by the family of all arithmetic progressions. It is a special case of the profinite topology on a group. This particular topological space was introduced by Furstenberg (1955) where it was used to prove the infinitude of primes.
In mathematics, more specifically general topology, the rational sequence topology is an example of a topology given to the set of real numbers, denoted R.