WikiMili The Free Encyclopedia

In the mathematical field of topology a **uniform property** or **uniform invariant** is a property of a uniform space which is invariant under uniform isomorphisms.

**Mathematics** includes the study of such topics as quantity, structure, space, and change.

In mathematics, **topology** is concerned with the properties of space that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing.

In the mathematical field of topology, a **uniform space** is a set with a **uniform structure**. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and uniform convergence.

Since uniform spaces come as topological spaces and uniform isomorphisms are homeomorphisms, every topological property of a uniform space is also a uniform property. This article is (mostly) concerned with uniform properties that are *not* topological properties.

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.

In the mathematical field of topology, a **homeomorphism**, **topological isomorphism**, or **bicontinuous function** is a 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. The word *homeomorphism* comes from the Greek words *ὅμοιος* (*homoios*) = similar or same and *μορφή* (*morphē*) = shape, form, introduced to mathematics by Henri Poincaré in 1895.

In topology and related areas of mathematics a **topological property** or **topological invariant** is a property of a topological space which is invariant 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.

**Separated**. A uniform space*X*is separated if the intersection of all entourages is equal to the diagonal in*X*×*X*. This is actually just a topological property, and equivalent to the condition that the underlying topological space is Hausdorff (or simply*T*_{0}since every uniform space is completely regular).**Complete**. A uniform space*X*is complete if every Cauchy net in*X*converges (i.e. has a limit point in*X*).**Totally bounded**(or**Precompact**). A uniform space*X*is totally bounded if for each entourage*E*⊂*X*×*X*there is a finite cover {*U*_{i}} of*X*such that*U*_{i}×*U*_{i}is contained in*E*for all*i*. Equivalently,*X*is totally bounded if for each entourage*E*there exists a finite subset {*x*_{i}} of*X*such that*X*is the union of all*E*[*x*_{i}]. In terms of uniform covers,*X*is totally bounded if every uniform cover has a finite subcover.**Compact**. A uniform space is compact if it is complete and totally bounded. Despite the definition given here, compactness is a topological property and so admits a purely topological description (every open cover has a finite subcover).**Uniformly connected**. A uniform space*X*is uniformly connected if every uniformly continuous function from*X*to a discrete uniform space is constant.**Uniformly disconnected**. A uniform space*X*is uniformly disconnected if it is not uniformly connected.

In topology and related branches of mathematics, a **Hausdorff space**, **separated space** or **T _{2} space** is a topological space where for any two distinct points there exists a neighbourhood of each which is disjoint from the neighbourhood of the other. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T

In mathematics, a **limit point** of a set *S* in a topological space *X* is a point *x* that can be "approximated" by points of *S* in the sense that every neighbourhood of *x* with respect to the topology on *X* also contains a point of *S* other than *x* itself. A limit point of a set *S* does not itself have to be an element of *S*.

In mathematics, a **cover** of a set is a collection of sets whose union contains as a subset. Formally, if

In mathematics, and 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 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 nonempty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces.

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 differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; this is the sense in which the product topology is "natural".

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 mathematics, a **topological group** is a group *G* together with a topology on *G* such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a topological structure. Thus, one may perform algebraic operations, because of the group structure, and one may talk about continuous functions, because of the topology.

In mathematics, a **topological vector space** is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, thereby admitting a notion of continuity. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.

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.

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 other in a certain sense. The discrete topology is the finest topology that can be given on a set, i.e., it defines all subsets as open sets. In particular, each singleton is an open set in the discrete topology.

In real analysis the **Heine–Borel theorem**, named after Eduard Heine and Émile Borel, states:

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, 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 topology, a **subbase** for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by some authors, and there are other useful equivalent formulations of the definition; these are discussed below.

In mathematics, an **amenable group** is a locally compact topological group *G* carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms of a finitely additive invariant measure on subsets of *G*, was introduced by John von Neumann in 1929 under the German name "messbar" in response to the Banach–Tarski paradox. In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun on "*mean*".

In mathematics, the **spectrum of a C*-algebra** or **dual of a C*-algebra***A*, denoted *Â*, is the set of unitary equivalence classes of irreducible *-representations of *A*. A *-representation π of *A* on a Hilbert space *H* is **irreducible** if, and only if, there is no closed subspace *K* different from *H* and {0} which is invariant under all operators π(*x*) with *x* ∈ *A*. We implicitly assume that irreducible representation means *non-null* irreducible representation, thus excluding trivial representations on one-dimensional spaces. As explained below, the spectrum *Â* is also naturally a topological space; this is similar to the notion of the spectrum of a ring.

In topology and related branches of mathematics, a **totally bounded space** is a space that can be covered by finitely many subsets of every fixed "size". The smaller the size fixed, the more subsets may be needed, but any specific size should require only finitely many subsets. A related notion is a **totally bounded set**, in which only a subset of the space needs to be covered. Every subset of a totally bounded space is a totally bounded set; but even if a space is not totally bounded, some of its subsets still will be.

- James, I. M. (1990).
*Introduction to Uniform Spaces*. Cambridge, UK: Cambridge University Press. ISBN 0-521-38620-9. - Willard, Stephen (1970).
*General Topology*. Reading, Massachusetts: Addison-Wesley. ISBN 0-486-43479-6.

The **International Standard Book Number** (**ISBN**) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.