Formal definition
Let and be topological spaces and let be a map from to that is continuous, closed, surjective and such that each fiber is compact relative to for each in . Then is known as a perfect map.
In mathematics, especially topology, a perfect map is a particular kind of continuous function between topological spaces. Perfect maps are weaker than homeomorphisms, but strong enough to preserve some topological properties such as local compactness that are not always preserved by continuous maps.
Let and be topological spaces and let be a map from to that is continuous, closed, surjective and such that each fiber is compact relative to for each in . Then is known as a perfect map.
In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable.
In mathematics and more specifically in 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.
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. For a list of terms specific to algebraic topology, see Glossary of algebraic topology.
In mathematics, topological groups are 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 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 named after the Russian mathematician Pavel Alexandroff. More precisely, let X be a topological space. Then the Alexandroff extension of X is a certain compact space X* together with an open embedding c : X → X* such that the complement of X in X* consists of a single point, typically denoted ∞. The map c is a Hausdorff compactification if and only if X is a locally compact, noncompact Hausdorff space. For such spaces the Alexandroff extension is called the one-point compactification or Alexandroff compactification. The advantages of the Alexandroff compactification lie in its simple, often geometrically meaningful structure and the fact that it is in a precise sense minimal among all compactifications; the disadvantage lies in the fact that it only gives a Hausdorff compactification on the class of locally compact, noncompact Hausdorff spaces, unlike the Stone–Čech compactification which exists for any topological 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.
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 topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local structure. If is a local homeomorphism, is said to be an étale space over Local homeomorphisms are used in the study of sheaves. Typical examples of local homeomorphisms are covering 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 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 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 composition of two continuous maps is again continuous, and the identity function is continuous. The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology.
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 mathematics, an LF-space, also written (LF)-space, is a topological vector space (TVS) X that is a locally convex inductive limit of a countable inductive system of Fréchet spaces. This means that X is a direct limit of a direct system in the category of locally convex topological vector spaces and each is a Fréchet space. The name LF stands for Limit of Fréchet spaces.
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 spaces that satisfy a very weak axiom of countability, and all first-countable spaces are sequential.
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.
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 union of those subspaces. It is also sometimes called the weak topology generated by the family of subspaces, a notion that is quite different from the notion of a weak topology generated by a set of maps.
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem, is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map.
In mathematics, specifically topology, a sequence covering map is any of a class of maps between topological spaces whose definitions all somehow relate sequences in the codomain with sequences in the domain. Examples include sequentially quotient maps, sequence coverings, 1-sequence coverings, and 2-sequence coverings. These classes of maps are closely related to sequential spaces. If the domain and/or codomain have certain additional topological properties then these definitions become equivalent to other well-known classes of maps, such as open maps or quotient maps, for example. In these situations, characterizations of such properties in terms of convergent sequences might provide benefits similar to those provided by, say for instance, the characterization of continuity in terms of sequential continuity or the characterization of compactness in terms of sequential compactness.
