Baire category theorem
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
Spectral space
In mathematics, a spectral space is a topological space that is homeomorphic to the spectrum of a commutative ring. It is sometimes also called a coherent space because of the connection to coherent t
Eberlein compactum
In mathematics an Eberlein compactum, studied by William Frederick Eberlein, is a compact topological space homeomorphic to a subset of a Banach space with the weak topology.Every compact metric space
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the rea
Cover (topology)
In mathematics, and more particularly in set theory, a cover (or covering) of a set is a collection of subsets of whose union is all of . More formally, if is an indexed family of subsets , then is a
Net (mathematics)
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function whose do
Order 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
First-countable space
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space is said to be first-countable if each point has
Initial topology
In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set with respect to a family of function
Volterra's function
In mathematics, Volterra's function, named for Vito Volterra, is a real-valued function V defined on the real line R with the following curious combination of properties:
* V is differentiable everyw
Comparison of topologies
In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies.
Sierpiński space
In mathematics, the Sierpiński space (or the connected two-point set) is a finite topological space with two points, only one of which is closed.It is the smallest example of a topological space which
Locally connected space
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.
Local property
In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some sufficiently small or arbitra
Rational sequence topology
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. To give R a topology means to say which subs
Cauchy space
In general topology and analysis, a Cauchy space is a generalization of metric spaces and uniform spaces for which the notion of Cauchy convergence still makes sense. Cauchy spaces were introduced by
Γ-space
In mathematics, a -space is a topological space that satisfies a certain a basic selection principle. An infinite cover of a topological space is an -cover if every finite subset of this space is cont
Axiom of countability
In mathematics, an axiom of countability is a property of certain mathematical objects that asserts the existence of a countable set with certain properties. Without such an axiom, such a set might no
Wallman compactification
In mathematics, the Wallman compactification, generally called Wallman–Shanin compactification is a compactification of T1 topological spaces that was constructed by .
Derived set (mathematics)
In mathematics, more specifically in point-set topology, the derived set of a subset of a topological space is the set of all limit points of It is usually denoted by The concept was first introduced
List of examples in general topology
This is a list of useful examples in general topology, a field of mathematics.
* Alexandrov topology
* Cantor space
* Co-kappa topology
* Cocountable topology
* Cofinite topology
* Compact-open
Appert topology
In general topology, a branch of mathematics, the Appert topology, named for Antoine Appert, is a topology on the set X = {1, 2, 3, ...} of positive integers.In the Appert topology, the open sets are
N-skeleton
In mathematics, particularly in algebraic topology, the n-skeleton of a topological space X presented as a simplicial complex (resp. CW complex) refers to the subspace Xn that is the union of the simp
Uniform continuity
In mathematics, a real function of real numbers is said to be uniformly continuous if there is a positive real number such that function values over any function domain interval of the size are as clo
Limit (mathematics)
In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential to calculus and mathematical analysis, and are used to
Point-finite collection
In mathematics, a collection of subsets of a topological space is said to be point-finite if every point of lies in only finitely many members of . A topological space in which every open cover admi
Ellis–Numakura lemma
In mathematics, the Ellis–Numakura lemma states that if S is a non-empty semigroup with a topology such that S is compact and the product is semi-continuous, then S has an idempotent element p, (that
Glossary of 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 topolo
Open and closed maps
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 i
Cut point
In topology, a cut-point is a point of a connected space such that its removal causes the resulting space to be disconnected. If removal of a point doesn't result in disconnected spaces, this point is
Upper topology
In mathematics, the upper topology on a partially ordered set X is the coarsest topology in which the closure of a singleton is the order section for each If is a partial order, the upper topology is
Baire set
In mathematics, more specifically in measure theory, the Baire sets form a σ-algebra of a topological space that avoids some of the pathological properties of Borel sets. There are several inequivalen
Alexandroff extension
In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is
Hilbert manifold
In mathematics, a Hilbert manifold is a manifold modeled on Hilbert spaces. Thus it is a separable Hausdorff space in which each point has a neighbourhood homeomorphic to an infinite dimensional Hilbe
Sequential space
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as
Half-disk topology
In mathematics, and particularly general topology, the half-disk topology is an example of a topology given to the set , given by all points in the plane such that . The set can be termed the closed u
Esenin-Volpin's theorem
In mathematics, Esenin-Volpin's theorem states that weight of an infinite compact dyadic space is the supremum of the weights of its points.It was introduced by Alexander Esenin-Volpin. It was general
Set-theoretic topology
In mathematics, set-theoretic topology is a subject that combines set theory and general topology. It focuses on topological questions that are independent of Zermelo–Fraenkel set theory (ZFC).
Product topology
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 dif
Denjoy–Riesz theorem
In topology, the Denjoy–Riesz theorem states that every compact set of totally disconnected points in the Euclidean plane can be covered by a continuous image of the unit interval, without self-inters
Pointclass
In the mathematical field of descriptive set theory, a pointclass is a collection of sets of points, where a point is ordinarily understood to be an element of some perfect Polish space. In practice,
Tychonoff cube
In mathematics, more specifically in general topology, the Tychonoff cube is the generalization of the unit cube from the product of a finite number of unit intervals to the product of an infinite, ev
Linear flow on the torus
In mathematics, especially in the area of mathematical analysis known as dynamical systems theory, a linear flow on the torus is a flow on the n-dimensional torus which is represented by the following
Local homeomorphism
In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure. If is a local h
Totally disconnected space
In topology and related branches of mathematics, a totally disconnected space is a topological space that is maximally disconnected, in the sense that it has no non-trivial connected subsets. In every
Accumulation point
In mathematics, a limit point, accumulation point, or cluster 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 res
Scott continuity
In mathematics, given two partially ordered sets P and Q, a function f: P → Q between them is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema. That is,
Separable space
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
Countably generated space
In mathematics, a topological space is called countably generated if the topology of is determined by the countable sets in a similar way as the topology of a sequential space (or a Fréchet space) is
Katětov–Tong insertion theorem
The Katětov–Tong insertion theorem is a theorem of point-set topology proved independently by Miroslav Katětov and Hing Tong in the 1950s. The theorem states the following: Let be a normal topological
Σ-compact space
In mathematics, a topological space is said to be σ-compact if it is the union of countably many compact subspaces. A space is said to be σ-locally compact if it is both σ-compact and (weakly) locally
Nested interval topology
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. T
Lexicographic order topology on the unit square
In general topology, the lexicographic ordering on the unit square (sometimes the dictionary order on the unit square) is a topology on the unit square S, i.e. on the set of points (x,y) in the plane
Axiomatic foundations of topological spaces
In the mathematical field of topology, a topological space is usually defined by declaring its open sets. However, this is not necessary, as there are many equivalent axiomatic foundations, each leadi
Coercive function
In mathematics, a coercive function is a function that "grows rapidly" at the extremes of the space on which it is defined. Depending on the contextdifferent exact definitions of this idea are in use.
Furstenberg's proof of the infinitude of primes
In mathematics, particularly in number theory, Hillel Furstenberg's proof of the infinitude of primes is a topological proof that the integers contain infinitely many prime numbers. When examined clos
Metrizable space
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space is said to be metrizable if there is a metr
Excisive triad
In topology, a branch of mathematics, an excisive triad is a triple of topological spaces such that A, B are subspaces of X and X is the union of the interior of A and the interior of B. Note B is not
Moore space (topology)
In mathematics, more specifically point-set topology, a Moore space is a developable regular Hausdorff space. That is, a topological space X is a Moore space if the following conditions hold:
* Any t
Baire space
In mathematics, a topological space is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.According to the Baire category theorem, compact Hausdor
Second-countable space
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space is second-countable if
Directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set together with a reflexive and transitive binary relation (that is, a preorder), with the additional property
Boundary (topology)
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
Mapping torus
In mathematics, the mapping torus in topology of a homeomorphism f of some topological space X to itself is a particular geometric construction with f. Take the cartesian product of X with a closed in
Compact-open topology
In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on functi
Proximity space
In topology, a proximity space, also called a nearness space, is an axiomatization of the intuitive notion of "nearness" that hold set-to-set, as opposed to the better known point-to-set notion that c
Completely metrizable space
In mathematics, a completely metrizable space (metrically topologically complete space) is a topological space (X, T) for which there exists at least one metric d on X such that (X, d) is a complete m
Pretopological space
In general topology, a pretopological space is a generalization of the concept of topological space. A pretopological space can be defined in terms of either filters or a preclosure operator. The simi
Filter (mathematics)
In mathematics, a filter or order filter is a special subset of a partially ordered set (poset). Filters appear in order and lattice theory, but can also be found in topology, from which they originat
Non-Hausdorff manifold
In geometry and topology, it is a usual axiom of a manifold to be a Hausdorff space. In general topology, this axiom is relaxed, and one studies non-Hausdorff manifolds: spaces locally homeomorphic to
General topology
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 topolo
Topological indistinguishability
In topology, two points of a topological space X are topologically indistinguishable if they have exactly the same neighborhoods. That is, if x and y are points in X, and Nx is the set of all neighbor
Exhaustion by compact sets
In mathematics, especially general topology and analysis, an exhaustion by compact sets of a topological space is a nested sequence of compact subsets of (i.e. ), such that is contained in the interio
Generic point
In algebraic geometry, a generic point P of an algebraic variety X is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost
Kuratowski–Ulam theorem
In mathematics, the Kuratowski–Ulam theorem, introduced by Kazimierz Kuratowski and Stanislaw Ulam, called also the Fubini theorem for category, is an analog of Fubini's theorem for arbitrary second c
Filters in topology
Filters in topology, a subfield of mathematics, can be used to study topological spaces and define all basic topological notions such a convergence, continuity, compactness, and more. Filters, which a
Embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object is said to be embedde
Fréchet–Urysohn space
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 a
Esakia space
In mathematics, Esakia spaces are special ordered topological spaces introduced and studied by in 1974. Esakia spaces play a fundamental role in the study of Heyting algebras, primarily by virtue of t
Polyadic space
In mathematics, a polyadic space is a topological space that is the image under a continuous function of a topological power of an Alexandroff one-point compactification of a discrete space.
Gδ space
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 b
Locally finite space
In the mathematical field of topology, a locally finite space is a topological space in which every point has a finite neighborhood, that is, an open neighborhood consisting of finitely many elements.
Saturated set
In mathematics, particularly in the subfields of set theory and topology, a set is said to be saturated with respect to a function if is a subset of 's domain and if whenever sends two points and to t
Adherent point
In mathematics, an adherent point (also closure point or point of closure or contact point) of a subset of a topological space is a point in such that every neighbourhood of (or equivalently, every op
Poset topology
In mathematics, the poset topology associated to a poset (S, ≤) is the Alexandrov topology (open sets are upper sets) on the poset of finite chains of (S, ≤), ordered by inclusion. Let V be a set of v
Thomae's function
Thomae's function is a real-valued function of a real variable that can be defined as: It is named after Carl Johannes Thomae, but has many other names: the popcorn function, the raindrop function, th
Discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each o
Polish space
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable
Development (topology)
In the mathematical field of topology, a development is a countable collection of open covers of a topological space that satisfies certain separation axioms. Let be a topological space. A development
Closed set
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 l
Open set
In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with eve
Irreducible component
In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component is an algeb
Zorich's theorem
In mathematical analysis, Zorich's theorem was proved by Vladimir A. Zorich in 1967. The result was conjectured by M. A. Lavrentev in 1938.
Gluing axiom
In mathematics, the gluing axiom is introduced to define what a sheaf on a topological space must satisfy, given that it is a presheaf, which is by definition a contravariant functor to a category whi
Compact embedding
In mathematics, the notion of being compactly embedded expresses the idea that one set or space is "well contained" inside another. There are versions of this concept appropriate to general topology a
Clopen set
In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of
End (topology)
In topology, a branch of mathematics, the ends of a topological space are, roughly speaking, the connected components of the "ideal boundary" of the space. That is, each end represents a topologically
Isolated point
In mathematics, a point x is called an isolated point of a subset S (in a topological space X) if x is an element of S and there exists a neighborhood of x which does not contain any other points of S
Nowhere dense set
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 (as defin
Stone space
In topology and related areas of mathematics, a Stone space, also known as a profinite space or profinite set, is a compact totally disconnected Hausdorff space. Stone spaces are named after Marshall
Subbase
In topology, a subbase (or subbasis, prebase, prebasis) for a topological space with topology is a subcollection of that generates in the sense that is the smallest topology containing A slightly diff
Stone functor
In mathematics, the Stone functor is a functor S: Topop → Bool, where Top is the category of topological spaces and Bool is the category of Boolean algebras and Boolean homomorphisms. It assigns to ea
Category of topological spaces
In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the comp
Curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a
Disjoint union (topology)
In general topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, or coproduct) of a family of topological spaces is a space
Neighbourhood (mathematics)
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Int
Locally finite collection
In the mathematical field of topology, local finiteness is a property of collections of subsets of a topological space. It is fundamental in the study of paracompactness and topological dimension. A c
Nodec space
In topology and related areas of mathematics, a topological space is a nodec space if every nowhere dense subset of is closed. This concept was introduced and studied by .
Parovicenko space
In mathematics, a Parovicenko space is a topological space similar to the space of non-isolated points of the Stone–Čech compactification of the integers.
Cocountable topology
The cocountable topology or countable complement topology on any set X consists of the empty set and all cocountable subsets of X, that is all sets whose complement in X is countable. It follows that
Cocompact embedding
In mathematics, cocompact embeddings are embeddings of normed vector spaces possessing a certain property similar to but weaker than compactness. Cocompactness has been in use in mathematical analysis
Interior (topology)
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 poi
Compact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space hav
P-space
In the mathematical field of topology, there are various notions of a P-space and of a p-space.
Nagata–Smirnov metrization theorem
The Nagata–Smirnov metrization theorem in topology characterizes when a topological space is metrizable. The theorem states that a topological space is metrizable if and only if it is regular, Hausdor
Subspace topology
In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a topology induced from that of X called the subspace topology (or the relat
Pointless topology
In mathematics, pointless topology (also called point-free or pointfree topology, or locale theory) is an approach to topology that avoids mentioning points, and in which the lattices of open sets are
Lawson topology
In mathematics and theoretical computer science the Lawson topology, named after Jimmie D. Lawson, is a topology on partially ordered sets used in the study of domain theory. The lower topology on a p
Closeness (mathematics)
Closeness is a basic concept in topology and related areas in mathematics. Intuitively we say two sets are close if they are arbitrarily near to each other. The concept can be defined naturally in a m
Topological space
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 s
Hedgehog space
In mathematics, a hedgehog space is a topological space consisting of a set of spines joined at a point. For any cardinal number , the -hedgehog space is formed by taking the disjoint union of real un
Counterexamples in Topology
Counterexamples in Topology (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr. In the process of working on problems like the metrization problem, topol
Remote point
In general topology, a remote point is a point that belongs to the Stone–Čech compactification of a Tychonoff space but that does not belong to the topological closure within of any nowhere dense subs
Unicoherent space
In mathematics, a unicoherent space is a topological space that is connected and in which the following property holds: For any closed, connected with , the intersection is connected. For example, any
Closure (topology)
In topology, 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
Heine–Borel theorem
In real analysis the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states: For a subset S of Euclidean space Rn, the following two statements are equivalent:
* S is closed and bounde
Stone–Čech compactification
In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a universal map from a topological space X to a compact
Gδ set
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 German with G for Gebiet (German: area, or
Sober space
In mathematics, a sober space is a topological space X such that every (nonempty) irreducible closed subset of X is the closure of exactly one point of X: that is, every irreducible closed subset has
Base (topology)
In mathematics, a base (or basis) for the topology τ of a topological space (X, τ) is a family of open subsets of X such that every open set of the topology is equal to the union of some sub-family of
Topological game
A topological game is an infinite game of perfect information played between two players on a topological space. Players choose objects with topological properties such as points, open sets, closed se
Quotient space (topology)
In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the or
Either–or topology
In mathematics, the either–or topology is a topological structure defined on the closed interval [−1, 1] by declaring a set open if it either does not contain {0} or does contain (−1, 1).
Banach–Mazur game
In general topology, set theory and game theory, a Banach–Mazur game is a topological game played by two players, trying to pin down elements in a set (space). The concept of a Banach–Mazur game is cl
Lindelöf space
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
Filter (set theory)
In mathematics, a filter on a set is a family of subsets such that: 1.
* and 2.
* if and ,then 3.
* If ,and ,then A filter on a set may be thought of as representing a "collection of large subsets"
Homotopical connectivity
In algebraic topology, homotopical connectivity is a property describing a topological space based on the dimension of its holes. In general, low homotopical connectivity indicates that the space has
Standard Borel space
In mathematics, a standard Borel space is the Borel space associated to a Polish space. Discounting Borel spaces of discrete Polish spaces, there is, up to isomorphism of measurable spaces, only one s
Cofiniteness
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 counta
Cylinder set
In mathematics, the cylinder sets form a basis of the product topology on a product of sets; they are also a generating family of the cylinder σ-algebra.
Dense set
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
Final topology
In general topology and related areas of mathematics, the final topology (or coinduced, strong, colimit, or inductive topology) on a set with respect to a family of functions from topological spaces i
Connected space
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 o
Integer broom topology
In general topology, a branch of mathematics, the integer broom topology is an example of a topology on the so-called integer broom space X.
S and L spaces
In mathematics, S-space is a regular topological space that is hereditarily separable but is not a Lindelöf space. L-space is a regular topological space that is hereditarily Lindelöf but not separabl
Coherent topology
In topology, a coherent topology is a topology that is uniquely determined by a family of subspaces. Loosely speaking, a topological space is coherent with a family of subspaces if it is a topological
Meagre set
In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense deta
Trivial topology
In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete o
Baire function
In mathematics, Baire functions are functions obtained from continuous functions by transfinite iteration of the operation of forming pointwise limits of sequences of functions. They were introduced b
Weak topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most
Finite topology
Finite topology is a mathematical concept which has several different meanings.
Regular open set
A subset of a topological space is called a regular open set if it is equal to the interior of its closure; expressed symbolically, if or, equivalently, if where and denote, respectively, the interior
Real-valued function
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a
Cellular space
A cellular space is a Hausdorff space that has the structure of a CW complex.
* v
* t
* e
Interlocking interval topology
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
Compactly generated space
In topology, a compactly generated space is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space X is compactly generated if it sa
Neighbourhood system
In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter for a point in a topological space is the collection of all neighbour
Completely uniformizable space
In mathematics, a topological space (X, T) is called completely uniformizable (or Dieudonné complete) if there exists at least one complete uniformity that induces the topology T. Some authors additio
Double origin topology
In mathematics, more specifically general topology, the double origin topology is an example of a topology given to the plane R2 with an extra point, say 0*, added. In this case, the double origin top
Finite intersection property
In general topology, a branch of mathematics, a non-empty family A of subsets of a set is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of is no