Properties
Let be a map between topological spaces.
In topology a branch of mathematics, a quasi-open map or quasi-interior map is a function which has similar properties to continuous maps. However, continuous maps and quasi-open maps are not related. [1]
A function f : X → Y between topological spaces X and Y is quasi-open if, for any non-empty open set U ⊆ X, the interior of f ('U) in Y is non-empty. [1] [2]
Let be a map between topological 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 topology and related branches of mathematics, a Hausdorff space ( HOWS-dorf, HOWZ-dorf), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters.
In the mathematical field of topology, a homeomorphism, also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same.
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 whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate.
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other.
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations are also continuous functions. Such a topology is called a vector topology and every topological vector space has a uniform topological structure, allowing a notion of uniform convergence and completeness. Some authors also require that the space is a Hausdorff space. One of the most widely studied categories of TVSs are locally convex topological vector spaces. This article focuses on TVSs that are not necessarily locally convex. Banach spaces, Hilbert spaces and Sobolev spaces are other well-known examples of TVSs.
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 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 original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonical projection map. In other words, a subset of a quotient space is open if and only if its preimage under the canonical projection map is open in the original topological space.
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.
In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each gives conditions when functions with closed graphs are necessarily continuous.
In topology, a topological space is called a compactly generated space or k-space if its topology is determined by compact spaces in a manner made precise below. There is in fact no commonly agreed upon definition for such spaces, as different authors use variations of the definition that are not exactly equivalent to each other. Also some authors include some separation axiom in the definition of one or both terms, and others don't.
In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spaces which is closed under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic to X possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.
In functional analysis and related areas of mathematics, a barrelled space is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a barrel in a topological vector space is a set that is convex, balanced, absorbing, and closed. Barrelled spaces are studied because a form of the Banach–Steinhaus theorem still holds for them. Barrelled spaces were introduced by Bourbaki (1950).
In mathematics, particularly in functional analysis, a webbed space is a topological vector space designed with the goal of allowing the results of the open mapping theorem and the closed graph theorem to hold for a wider class of linear maps whose codomains are webbed spaces. A space is called webbed if there exists a collection of sets, called a web that satisfies certain properties. Webs were first investigated by de Wilde.
In mathematics, particularly in functional analysis and topology, closed graph is a property of functions. A function f : X → Y between topological spaces has a closed graph if its graph is a closed subset of the product space X × Y. A related property is open graph.
A locally convex topological vector space (TVS) is B-complete or a Ptak space if every subspace is closed in the weak-* topology on whenever is closed in for each equicontinuous subset .
In functional analysis, a topological homomorphism or simply homomorphism is the analog of homomorphisms for the category of topological vector spaces (TVSs). This concept is of considerable importance in functional analysis and the famous open mapping theorem gives a sufficient condition for a continuous linear map between Fréchet spaces to be a topological homomorphism.
In mathematics, particularly in functional analysis and topology, the closed graph theorem is a result connecting the continuity of certain kinds of functions to a topological property of their graph. In its most elementary form, the closed graph theorem states that a linear function between two Banach spaces is continuous if and only if the graph of that function is closed.
In functional analysis and related areas of mathematics, an almost open map between topological spaces is a map that satisfies a condition similar to, but weaker than, the condition of being an open map. As described below, for certain broad categories of topological vector spaces, all surjective linear operators are necessarily almost open.
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