Tychonoff space

Suppose that X is a topological space. X is a completely regular space iff, given any closed set F and any point x that does not belong to F, there is a continuous function f from X to the real line R such that f(x) is 0 and f(y) is 1 for every y in F. In fancier terms, this condition says that x and F can be separated by a function. X is a Tychonoff space, or T3½ space, or Tπ space, or completely T3 space if and only i ...

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Tychonoff space, Tychonoff space - Definitions, Tychonoff space - Examples and counterexamples, Tychonoff space - Properties

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In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniform continuity and uniform convergence. The conceptual difference between uniform and topological structures is that in a uniform space, you can formalize the idea that "x is as close to a as y is to b", while in a topological space you can only formalize "x ...

Including:

• Uniform space - History
• Uniform space - Definition
• Uniform space - Entourage definition
• Uniform space - Uniform cover definition
• Uniform space - Pseudometrics definition
• Uniform space - Intuition
• Uniform space - Examples
• Uniform space - Uniformly continuous functions
• Uniform space - Topology of uniform spaces
• Uniform space - Completeness

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In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. For example, in R, the closed unit interval [0, 1] is compact, but the set of integers Z is not (it is not bounded) and neither is the half-open interval [0, 1) (it is not closed). A more modern approach is to call a topological space compact if all its open covers have a finite subcover. The Heine–Borel theorem affirms that this coincides with ...

Including:

• Compact space - History and motivation
• Compact space - Definitions
• Compact space - Compactness of subsets of Rⁿ
• Compact space - Compactness of topological spaces
• Compact space - Examples of compact spaces
• Compact space - Theorems
• Compact space - Other forms of compactness

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In mathematics, the axiom of choice, or AC, is an axiom of set theory. It was formulated in 1904 by Ernst Zermelo. While it was originally controversial, it is now used without embarrassment by most mathematicians. However, there are still schools of mathematical thought, primarily within set theory, that either reject the axiom of choice, or even investigate consequences of axioms inconsistent with AC. Intuitively speaking, AC says that given a collection of bins, each containing at least one object, then exactly one ob ...

Including:

• Axiom of choice - Statement
• Axiom of choice - Usage
• Axiom of choice - Independence of AC
• Axiom of choice - Weaker forms of AC
• Axiom of choice - Results requiring AC or weaker forms
• Axiom of choice - Results requiring ¬AC
• Axiom of choice - Results requiring choice in intuitionistic logic though not classically
• Axiom of choice - Quotes

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In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another. Such functors are ubiquitous in mathematics. Adjoint functors are studied in a branch of mathematics known as category theory. Like much of category theory, the general notion of adjoint functors arises at an abstract level beyond the everyday usage of mathematicians. Adjoint functors can be considered from several different points of view. This article starts with a number of introductory sections considering some ...

Including:

• Adjoint functors - Motivation
• Adjoint functors - Ubiquity of adjoint functors
• Adjoint functors - Deep problems formulated with adjoint functors
• Adjoint functors - Adjoint functors as solving optimization problems
• Adjoint functors - The case of partial orders
• Adjoint functors - Formal definitions
• Adjoint functors - Examples
• Adjoint functors - Properties
• Adjoint functors - Uniqueness of adjoints
• Adjoint functors - Relation to universal constructions
• Adjoint functors - Characterization via unit and co-unit
• Adjoint functors - Adjoints preserve certain limits
• Adjoint functors - Additivity
• Adjoint functors - Composition
• Adjoint functors - Adjoint pairs extend equivalences
• Adjoint functors - General existence theorem

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Locally compact space - Compact Hausdorff spaces. Every compact Hausdorff space is also locally compact, and many examples of compact spaces may be found in the article Compact space. Here we mention only: the unit interval [0,1]; any closed topological manifold; the Cantor set; the Hilbert cube. Locally compact spa ...

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Locally compact space, Locally compact space - Examples and nonexamples, Locally compact space - Compact Hausdorff spaces, Locally compact space - Locally compact Hausdorff spaces that are not compact, Locally compact space - Hausdorff spaces that are not locally compact, Locally compact space - Facts about locally compact Hausdorff spaces, Locally compact space - The point at infinity, Locally compact space - Locally compact groups, Locally compact space - Non-Hausdorff spaces

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Until the late 19th century, the axiom of choice was often used implicitly. For example, after having established that the set X contains only non-empty sets, a mathematician might have said "let F(s) be one of the members of s for all s in X." In general, it is impossible to prove that F exists without the axiom of choice, but this seems to have gone unnoticed until Zermelo. Not every situation requires the axiom of choice. For finite sets X, the axiom of choice follows from the other ...

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Axiom of choice, Axiom of choice - Statement, Axiom of choice - Usage, Axiom of choice - Independence of AC, Axiom of choice - Weaker forms of AC, Axiom of choice - Results requiring AC or weaker forms, Axiom of choice - Results requiring ¬AC, Axiom of choice - Results requiring choice in intuitionistic logic though not classically, Axiom of choice - Quotes

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List of general topology topics - Compactness and countability. Compact space Relatively compact subspace Heine-Borel theorem Tychonoff's theorem Finite intersection property Compactification Measure of non-compactness Paracompact space Locally compact space Compactly generated space Axiom of countability First-countable space Second-countable space Separable space Lindel ...

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List of general topology topics, List of general topology topics - Basic concepts, List of general topology topics - Limits, List of general topology topics - Topological properties, List of general topology topics - Compactness and countability, List of general topology topics - Connectedness, List of general topology topics - Separation axioms, List of general topology topics - Topological constructions, List of general topology topics - Examples, List of general topology topics - Uniform spaces, List of general topology topics - Metric spaces, List of general topology topics - Topology and order theory, List of general topology topics - Descriptive set theory, List of general topology topics - Dimension theory, List of general topology topics - Topological algebra, List of general topology topics - Combinatorial topology, List of general topology topics - Foundations of algebraic topology

Read more here: » List of general topology topics: Encyclopedia II - List of general topology topics - Topological properties

Adjoint functors - Ubiquity of adjoint functors. The idea of an adjoint functor was formulated by Daniel Kan in 1958. Like many of the concepts in category theory, it was suggested by the needs of homological algebra, which was at the time devoted to computations. Those faced with giving tidy, systematic presentations of the subject would have noticed relations such as Hom(F(X), Y< ...

See also:

Adjoint functors, Adjoint functors - Motivation, Adjoint functors - Ubiquity of adjoint functors, Adjoint functors - Deep problems formulated with adjoint functors, Adjoint functors - Adjoint functors as solving optimization problems, Adjoint functors - The case of partial orders, Adjoint functors - Formal definitions, Adjoint functors - Examples, Adjoint functors - Properties, Adjoint functors - Uniqueness of adjoints, Adjoint functors - Relation to universal constructions, Adjoint functors - Characterization via unit and co-unit, Adjoint functors - Adjoints preserve certain limits, Adjoint functors - Additivity, Adjoint functors - Composition, Adjoint functors - Adjoint pairs extend equivalences, Adjoint functors - General existence theorem

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The following is an intuitive way of understanding the hyperreal numbers. The approach taken here is very close to the one in the book by Goldblatt (see the references below). Recall that the sequences converging to zero are sometimes called infinitely small. These are almost the infinitesimals in a sense, the true infinitesimals are the classes of sequences that contain a sequence converging to zero. Let us see where these classes come from. Consider first the sequences of real numbers. They form a ring, that is, one can multiply add and su ...

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Hyperreal number, Hyperreal number - The transfer principle, Hyperreal number - The ultrapower construction, Hyperreal number - An intuitive approach to the ultrapower construction, Hyperreal number - Infinitesimal and infinite numbers, Hyperreal number - Hyperreal fields

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Uniform space - Entourage definition. A uniform space (X,Φ) is a set X equipped with a nonempty family of subsets of the Cartesian product X × X (Φ is called the uniform structure of X and its elements entourages (French:neighborhoods) or surroundings) with the following properties if U is in Φ, then U contains the diagonal { (x, x) : x in X }. if U is in Φ and V i ...

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Uniform space, Uniform space - History, Uniform space - Definition, Uniform space - Entourage definition, Uniform space - Uniform cover definition, Uniform space - Pseudometrics definition, Uniform space - Intuition, Uniform space - Examples, Uniform space - Uniformly continuous functions, Uniform space - Topology of uniform spaces, Uniform space - Completeness

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Suppose that X is a topological space. X is a normal space if, given any disjoint closed sets E and F, there are a neighbourhood U of E and a neighbourhood V of F that are also disjoint. In fancier terms, this condition says that E and F can be separated by neighbourhoods. X is a T4 s ...

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Normal space, Normal space - Definitions, Normal space - Examples of normal spaces, Normal space - Examples of non-normal spaces, Normal space - Properties, Normal space - Relationships to other separation axioms

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The term compact was introduced by Fréchet in 1906. It has been recognized for a long time that a property like compactness was needed to prove many useful results. At one time, when primarily metric spaces were studied, compact was taken to mean sequentially compact (every sequence has a convergent subsequence). The definition based on open coverings has surpassed it by allowing many useful results that could be proven about metric ...

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Compact space, Compact space - History and motivation, Compact space - Definitions, Compact space - Compactness of subsets of Rⁿ, Compact space - Compactness of topological spaces, Compact space - Examples of compact spaces, Compact space - Theorems, Compact space - Other forms of compactness

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Interestingly, in various varieties of constructive logic (in particular, intuitionistic logic) in which the law of excluded middle is not assumed, the assumption of the axiom of choice is sufficient to obtain the law of excluded middle as a theorem. To see this, for any proposition let be the set and let be the set (see Set-builder notation). By the axiom of choice, there will exist a choice function for the set (note that, although the axiom of choice isn't classically required in order to obtain choice functions for finite sets, it ...

See also:

Axiom of choice, Axiom of choice - Statement, Axiom of choice - Usage, Axiom of choice - Independence of AC, Axiom of choice - Weaker forms of AC, Axiom of choice - Results requiring AC or weaker forms, Axiom of choice - Results requiring ¬AC, Axiom of choice - Results requiring choice in intuitionistic logic, though not classically, Axiom of choice - Quotes

Read more here: » Axiom of choice: Encyclopedia II - Axiom of choice - Results requiring choice in intuitionistic logic, though not classically

A hyperreal number r is called infinitesimal if it is smaller than every positive real number and bigger than every negative real number. Zero is an infinitesimal, but non-zero infinitesimals also exist: take for instance the class of the sequence (1, 1/2, 1/3, 1/4, 1/5, 1/6, ...) (this works because the ultrafilter U contains all index sets whose complement is finite). A hyperreal number x is called finite (or limited by some authors) if there exists a natural number n such that -n< ...

See also:

Hyperreal number, Hyperreal number - The transfer principle, Hyperreal number - The ultrapower construction, Hyperreal number - An intuitive approach to the ultrapower construction, Hyperreal number - Infinitesimal and infinite numbers, Hyperreal number - Hyperreal fields

Read more here: » Hyperreal number: Encyclopedia II - Hyperreal number - Infinitesimal and infinite numbers

The main significance of normal spaces lies in the fact that they admit "enough" continuous real-valued functions, as expressed by the following theorems valid for any normal space X: The Urysohn's lemma: If A and B are two disjoint closed subsets of X, then there exists a continuous function f from X to the real line R such that f(x) = 0 for all x in A and f(x) = 1 for all x in B. In fact, we can take the values of f to be ...

See also:

Normal space, Normal space - Definitions, Normal space - Examples of normal spaces, Normal space - Examples of non-normal spaces, Normal space - Properties, Normal space - Relationships to other separation axioms

Read more here: » Normal space: Encyclopedia II - Normal space - Properties

An important example of a non-normal topology is given by the Zariski topology on an algebraic variety or on the spectrum of a ring, which is used in algebraic geometry. A non-normal space of some relevance to analysis is the topological vector space of all functions from the real line R to itself, with the topology of pointwise convergence. More generally, a theorem of A. H. Stone states that the product of ...

See also:

Normal space, Normal space - Definitions, Normal space - Examples of normal spaces, Normal space - Examples of non-normal spaces, Normal space - Properties, Normal space - Relationships to other separation axioms

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Most spaces encountered in mathematical analysis are normal Hausdorff spaces, or at least normal regular spaces: All metric spaces (and hence all metrizable spaces) are perfectly normal Hausdorff; All pseudometric spaces (and hence all pseudometrisable spaces) are perfectly normal regular, although not in general Hausdorff; All compact Hausdorff spaces are normal; In particular, the Stone-Cech compactification of a Tychonoff space is normal Hausdorff; Generalizing the above examples, all parac ...

See also:

Normal space, Normal space - Definitions, Normal space - Examples of normal spaces, Normal space - Examples of non-normal spaces, Normal space - Properties, Normal space - Relationships to other separation axioms

Read more here: » Normal space: Encyclopedia II - Normal space - Examples of normal spaces

As mentioned in the previous section, any compact Hausdorff space is also locally compact, and any locally compact Hausdorff space is in fact a Tychonoff space. Every locally compact Hausdorff space is a Baire space. That is, the conclusion of the Baire category theorem holds: the interior of every union of countably many nowhere dense subsets is empty. A subspace X of a locally compact Hausdorff space Y is locally compact if and only if X can be written as the set-theoretic difference of two closed subsets ...

See also:

Locally compact space, Locally compact space - Examples and nonexamples, Locally compact space - Compact Hausdorff spaces, Locally compact space - Locally compact Hausdorff spaces that are not compact, Locally compact space - Hausdorff spaces that are not locally compact, Locally compact space - Facts about locally compact Hausdorff spaces, Locally compact space - The point at infinity, Locally compact space - Locally compact groups, Locally compact space - Non-Hausdorff spaces

Read more here: » Locally compact space: Encyclopedia II - Locally compact space - Facts about locally compact Hausdorff spaces

Every metric space (M, d) can be considered as a uniform space by defining a subset V of M × M to be an entourage if and only if there exists an ε > 0 such that for all x, y in M with d(x, y) < ε we have (x, y) in V. This uniform structure on M generates the usual topology on M. Using metrics, a simple example of distinct uniform structures with coinciding topologies can be constructed. For instance, let dSee also:

Uniform space, Uniform space - History, Uniform space - Definition, Uniform space - Entourage definition, Uniform space - Uniform cover definition, Uniform space - Pseudometrics definition, Uniform space - Intuition, Uniform space - Examples, Uniform space - Uniformly continuous functions, Uniform space - Topology of uniform spaces, Uniform space - Completeness

Read more here: » Uniform space: Encyclopedia II - Uniform space - Examples