Английская Википедия:Axiomatic foundations of topological spaces

In the mathematical field of topology, a topological space is usually defined by declaring its open sets.Шаблон:Sfnm However, this is not necessary, as there are many equivalent axiomatic foundations, each leading to exactly the same concept. For instance, a topological space determines a class of closed sets, of closure and interior operators, and of convergence of various types of objects. Each of these can instead be taken as the primary class of objects, with all of the others (including the class of open sets) directly determined from that new starting point. For example, in Kazimierz Kuratowski's well-known textbook on point-set topology, a topological space is defined as a set together with a certain type of "closure operator," and all other concepts are derived therefrom.Шаблон:Sfnm Likewise, the neighborhood-based axioms (in the context of Hausdorff spaces) can be retraced to Felix Hausdorff's original definition of a topological space in Grundzüge der Mengenlehre.Шаблон:Citation needed

Many different textbooks use many different inter-dependences of concepts to develop point-set topology. The result is always the same collection of objects: open sets, closed sets, and so on. For many practical purposes, the question of which foundation is chosen is irrelevant, as long as the meaning and interrelation between objects (many of which are given in this article), which are the same regardless of choice of development, are understood. However, there are cases where it can be useful to have flexibility. For instance, there are various natural notions of convergence of measures, and it is not immediately clear whether they arise from a topological structure or not. Such questions are greatly clarified by the topological axioms based on convergence.

Standard definitions via open setsШаблон:Anchor

A topological space is a set <math>X</math> together with a collection <math>S</math> of subsets of <math>X</math> satisfying:Шаблон:Sfnm

The empty set and <math>X</math> are in <math>S.</math>
The union of any collection of sets in <math>S</math> is also in <math>S.</math>
The intersection of any pair of sets in <math>S</math> is also in <math>S.</math> Equivalently, the intersection of any finite collection of sets in <math>S</math> is also in <math>S.</math>

Given a topological space <math>(X, S),</math> one refers to the elements of <math>S</math> as the open sets of <math>X,</math> and it is common only to refer to <math>S</math> in this way, or by the label topology. Then one makes the following secondary definitions:

Given a second topological space <math>Y,</math> a function <math>f : X \to Y</math> is said to be continuous if and only if for every open subset <math>U</math> of <math>Y,</math> one has that <math>f^{-1}(U)</math> is an open subset of <math>X.</math>Шаблон:Sfnm
A subset <math>C</math> of <math>X</math> is closed if and only if its complement <math>X \setminus C</math> is open.Шаблон:Sfnm
Given a subset <math>A</math> of <math>X,</math> the closure is the set of all points such that any open set containing such a point must intersect <math>A.</math>Шаблон:Sfnm
Given a subset <math>A</math> of <math>X,</math> the interior is the union of all open sets contained in <math>A.</math>Шаблон:Sfnm
Given an element <math>x</math> of <math>X,</math> one says that a subset <math>A</math> is a neighborhood of <math>x</math> if and only if <math>x</math> is contained in an open subset of <math>X</math> which is also a subset of <math>A.</math>Шаблон:Sfnm Some textbooks use "neighborhood of <math>x</math>" to instead refer to an open set containing <math>x.</math>Шаблон:Sfnm
One says that a net converges to a point <math>x</math> of <math>X</math> if for any open set <math>U</math> containing <math>x,</math> the net is eventually contained in <math>U.</math>Шаблон:Sfnm
Given a set <math>X,</math> a filter is a collection of nonempty subsets of <math>X</math> that is closed under finite intersection and under supersets.Шаблон:Sfnm Some textbooks allow a filter to contain the empty set, and reserve the name "proper filter" for the case in which it is excluded.Шаблон:Sfnm A topology on <math>X</math> defines a notion of a filter converging to a point <math>x</math> of <math>X,</math> by requiring that any open set <math>U</math> containing <math>x</math> is an element of the filter.Шаблон:Sfnm
Given a set <math>X,</math> a filterbase is a collection of nonempty subsets such that every two subsets intersect nontrivially and contain a third subset in the intersection.Шаблон:Sfnm Given a topology on <math>X,</math> one says that a filterbase converges to a point <math>x</math> if every neighborhood of <math>x</math> contains some element of the filterbase.Шаблон:Sfnm

Definition via closed setsШаблон:Anchor

Let <math>X</math> be a topological space. According to De Morgan's laws, the collection <math>T</math> of closed sets satisfies the following properties:Шаблон:Sfnm

The empty set and <math>X</math> are elements of <math>T</math>
The intersection of any collection of sets in <math>T</math> is also in <math>T.</math>
The union of any pair of sets in <math>T</math> is also in <math>T.</math>

Now suppose that <math>X</math> is only a set. Given any collection <math>T</math> of subsets of <math>X</math> which satisfy the above axioms, the corresponding set <math>\{U : X \setminus U \in T\}</math> is a topology on <math>X,</math> and it is the only topology on <math>X</math> for which <math>T</math> is the corresponding collection of closed sets.Шаблон:Sfnm This is to say that a topology can be defined by declaring the closed sets. As such, one can rephrase all definitions to be in terms of closed sets:

Given a second topological space <math>Y,</math> a function <math>f : X \to Y</math> is continuous if and only if for every closed subset <math>U</math> of <math>Y,</math> the set <math>f^{-1}(U)</math> is closed as a subset of <math>X.</math>Шаблон:Sfnm
a subset <math>C</math> of <math>X</math> is open if and only if its complement <math>X \setminus C</math> is closed.Шаблон:Sfnm
given a subset <math>A</math> of <math>X,</math> the closure is the intersection of all closed sets containing <math>A.</math>Шаблон:Sfnm
given a subset <math>A</math> of <math>X,</math> the interior is the complement of the intersection of all closed sets containing <math>X \setminus A.</math>

Definition via closure operatorsШаблон:Anchor

Given a topological space <math>X,</math> the closure can be considered as a map <math>\wp(X) \to \wp(X),</math> where <math>\wp(X)</math> denotes the power set of <math>X.</math> One has the following Kuratowski closure axioms:Шаблон:Sfnm

<math>A \subseteq \operatorname{cl}(A)</math>
<math>\operatorname{cl}(\operatorname{cl}(A)) = \operatorname{cl}(A)</math>
<math>\operatorname{cl}(A \cup B) = \operatorname{cl}(A) \cup \operatorname{cl}(B)</math>
<math>\operatorname{cl}(\varnothing) = \varnothing</math>

If <math>X</math> is a set equipped with a mapping satisfying the above properties, then the set of all possible outputs of cl satisfies the previous axioms for closed sets, and hence defines a topology; it is the unique topology whose associated closure operator coincides with the given cl.Шаблон:Sfnm As before, it follows that on a topological space <math>X,</math> all definitions can be phrased in terms of the closure operator:

Given a second topological space <math>Y,</math> a function <math>f : X \to Y</math> is continuous if and only if for every subset <math>A</math> of <math>X,</math> one has that the set <math>f(\operatorname{cl}(A))</math> is a subset of <math>\operatorname{cl}(f(A)).</math>Шаблон:Sfnm
A subset <math>A</math> of <math>X</math> is open if and only if <math>\operatorname{cl}(X \setminus A) = X \setminus A.</math>Шаблон:Sfnm
A subset <math>C</math> of <math>X</math> is closed if and only if <math>\operatorname{cl}(C) = C.</math>Шаблон:Sfnm
Given a subset <math>A</math> of <math>X,</math> the interior is the complement of <math>\operatorname{cl}(X \setminus A).</math>Шаблон:Sfnm

Definition via interior operatorsШаблон:Anchor

Given a topological space <math>X,</math> the interior can be considered as a map <math>\wp(X) \to \wp(X),</math> where <math>\wp(X)</math> denotes the power set of <math>X.</math> It satisfies the following conditions:Шаблон:Sfnm

<math>\operatorname{int}(A) \subseteq A</math>
<math>\operatorname{int}(\operatorname{int}(A)) = \operatorname{int}(A)</math>
<math>\operatorname{int}(A \cap B) = \operatorname{int}(A) \cap \operatorname{int}(B)</math>
<math>\operatorname{int}(X) = X</math>

If <math>X</math> is a set equipped with a mapping satisfying the above properties, then the set of all possible outputs of int satisfies the previous axioms for open sets, and hence defines a topology; it is the unique topology whose associated interior operator coincides with the given int.Шаблон:Sfnm It follows that on a topological space <math>X,</math> all definitions can be phrased in terms of the interior operator, for instance:

Given topological spaces <math>X</math> and <math>Y,</math> a function <math>f : X \to Y</math> is continuous if and only if for every subset <math>B</math> of <math>Y,</math> one has that the set <math>f^{-1}(\operatorname{int}(B))</math> is a subset of <math>\operatorname{int}(f^{-1}(B)).</math>Шаблон:Sfnm
A set is open if and only if it equals its interior.Шаблон:Sfnm
The closure of a set is the complement of the interior of its complement.Шаблон:Sfnm

Definition via neighbourhoodsШаблон:Anchor

Шаблон:See also

Recall that this article follows the convention that a neighborhood is not necessarily open. In a topological space, one has the following facts:Шаблон:Sfn

If <math>U</math> is a neighborhood of <math>x</math> then <math>x</math> is an element of <math>U.</math>
The intersection of two neighborhoods of <math>x</math> is a neighborhood of <math>x.</math> Equivalently, the intersection of finitely many neighborhoods of <math>x</math> is a neighborhood of <math>x.</math>
If <math>V</math> contains a neighborhood of <math>x,</math> then <math>V</math> is a neighborhood of <math>x.</math>
If <math>U</math> is a neighborhood of <math>x,</math> then there exists a neighborhood <math>V</math> of <math>x</math> such that <math>U</math> is a neighborhood of each point of <math>V</math>.

If <math>X</math> is a set and one declares a nonempty collection of neighborhoods for every point of <math>X,</math> satisfying the above conditions, then a topology is defined by declaring a set to be open if and only if it is a neighborhood of each of its points; it is the unique topology whose associated system of neighborhoods is as given.Шаблон:Sfn It follows that on a topological space <math>X,</math> all definitions can be phrased in terms of neighborhoods:

Given another topological space <math>Y,</math> a map <math>f : X \to Y</math> is continuous if and only for every element <math>x</math> of <math>X</math> and every neighborhood <math>B</math> of <math>f(x),</math> the preimage <math>f^{-1}(B)</math> is a neighborhood of <math>x.</math>Шаблон:Sfnm
A subset of <math>X</math> is open if and only if it is a neighborhood of each of its points.
Given a subset <math>A</math> of <math>X,</math> the interior is the collection of all elements <math>x</math> of <math>X</math> such that <math>A</math> is a neighbourhood of <math>x</math>.
Given a subset <math>A</math> of <math>X,</math> the closure is the collection of all elements <math>x</math> of <math>X</math> such that every neighborhood of <math>x</math> intersects <math>A.</math>Шаблон:Sfnm

Definition via convergence of netsШаблон:Anchor

Шаблон:See also

Convergence of nets satisfies the following properties:Шаблон:Sfnm Шаблон:Sfn

Every constant net converges to itself.
Every subnet of a convergent net converges to the same limits.
If a net does not converge to a point <math>x</math> then there is a subnet such that no further subnet converges to <math>x.</math> Equivalently, if <math>x_{\bull}</math> is a net such that every one of its subnets has a sub-subnet that converges to a point <math>x,</math> then <math>x_{\bull}</math> converges to <math>x.</math>
Шаблон:Visible anchor/Convergence of iterated limits. If <math>\left(x_a\right)_{a \in A} \to x</math> in <math>X</math> and for every index <math>a \in A,</math> <math>\left(x_a^i\right)_{i \in I_a}</math> is a net that converges to <math>x_a</math> in <math>X,</math> then there exists a diagonal (sub)net of <math>\left(x_a^i\right)_{a \in A, i \in I_a}</math> that converges to <math>x.</math>
- A Шаблон:Em refers to any subnet of <math>\left(x_a^i\right)_{a \in A, i \in I_a}.</math>
- The notation <math>\left(x_a^i\right)_{a \in A, i \in I_a}</math> denotes the net defined by <math>(a, i) \mapsto x_a^i</math> whose domain is the set <math>{\textstyle\bigcup\limits_{a \in A}} A \times I_a</math> ordered lexicographically first by <math>A</math> and then by <math>I_a;</math>Шаблон:Sfn explicitly, given any two pairs <math>(a_1, i_1), \left(a_2, i_2\right) \in {\textstyle\bigcup\limits_{a \in A}} A \times I_a,</math> declare that <math>(a_1, i_1) \leq \left(a_2, i_2\right)</math> holds if and only if both (1) <math>a_1 \leq a_2,</math> and also (2) if <math>a_1 = a_2</math> then <math>i_1 \leq i_2.</math>

If <math>X</math> is a set, then given a notion of net convergence (telling what nets converge to what pointsШаблон:Sfn) satisfying the above four axioms, a closure operator on <math>X</math> is defined by sending any given set <math>A</math> to the set of all limits of all nets valued in <math>A;</math> the corresponding topology is the unique topology inducing the given convergences of nets to points.Шаблон:Sfnm

Given a subset <math>A \subseteq X</math> of a topological space <math>X:</math>

<math>A</math> is open in <math>X</math> if and only if every net converging to an element of <math>A</math> is eventually contained in <math>A.</math>
the closure of <math>A</math> in <math>X</math> is the set of all limits of all convergent nets valued in <math>A.</math>Шаблон:Sfnm Шаблон:Sfn
<math>A</math> is closed in <math>X</math> if and only if there does not exist a net in <math>A</math> that converges to an element of the complement <math>X \setminus A.</math>Шаблон:Sfnm A subset <math>A \subseteq X</math> is closed in <math>X</math> if and only if every limit point of every convergent net in <math>A</math> necessarily belongs to <math>A.</math>Шаблон:Sfn

A function <math>f : X \to Y</math> between two topological spaces is continuous if and only if for every <math>x \in X</math> and every net <math>x_{\bull}</math> in <math>X</math> that converges to <math>x</math> in <math>X,</math> the net <math>f\left(x_{\bull}\right)</math>^{[note 1]} converges to <math>f(x)</math> in <math>Y.</math>Шаблон:Sfnm

Definition via convergence of filtersШаблон:Anchor

Шаблон:See also

A topology can also be defined on a set by declaring which filters converge to which points.Шаблон:Citation needed One has the following characterizations of standard objects in terms of filters and prefilters (also known as filterbases):

Given a second topological space <math>Y,</math> a function <math>f : X \to Y</math> is continuous if and only if it preserves convergence of prefilters.Шаблон:Sfnm
A subset <math>A</math> of <math>X</math> is open if and only if every filter converging to an element of <math>A</math> contains <math>A.</math>Шаблон:Sfnm
A subset <math>A</math> of <math>X</math> is closed if and only if there does not exist a prefilter on <math>A</math> which converges to a point in the complement <math>X \setminus A.</math>Шаблон:Sfnm
Given a subset <math>A</math> of <math>X,</math> the closure consists of all points <math>x</math> for which there is a prefilter on <math>A</math> converging to <math>x.</math>Шаблон:Sfnm
A subset <math>A</math> of <math>X</math> is a neighborhood of <math>x</math> if and only if it is an element of every filter converging to <math>x.</math>Шаблон:Sfnm

Citations

Шаблон:Reflist

Notes Шаблон:Reflist

References

Ошибка цитирования Для существующих тегов <ref> группы «note» не найдено соответствующего тега <references group="note"/>

[note 1]

Партнерские ресурсы
Криптовалюты	Обмен криптовалют - www.bestchange.ru Криптовалютная биржа CoinEx Криптовалютная биржа Binance HIVE OS - операционная система для майнинга e4pool - Мультивалютный пул для майнинга.
Магазины	AliExpress — глобальная виртуальная (в Интернете) торговая площадка, предоставляющая возможность покупать товары производителей из КНР; computeruniverse.net - Интернет-магазин компьютеров(Промо код 5 Евро на первую покупку:FWWC3ZKQ);
Хостинг	DigitalOcean - американский провайдер облачных инфраструктур, с главным офисом в Нью-Йорке и с центрами обработки данных по всему миру;
Разное	Викиум - Онлайн-тренажер для мозга Like Центр - Центр поддержки и развития предпринимательства. Gamersbay - лучший магазин по бустингу для World of Warcraft. Ноотропы OmniMind N°1 - Усиливает мозговую активность. Повышает мотивацию. Улучшает память. Санкт-Петербургская школа телевидения - это федеральная сеть образовательных центров, которая имеет филиалы в 37 городах России. Lingualeo.com — интерактивный онлайн-сервис для изучения и практики английского языка в увлекательной игровой форме. Junyschool (Джунискул) – международная школа программирования и дизайна для детей и подростков от 5 до 17 лет, где ученики осваивают компьютерную грамотность, развивают алгоритмическое и креативное мышление, изучают основы программирования и компьютерной графики, создают собственные проекты: игры, сайты, программы, приложения, анимации, 3D-модели, монтируют видео. Умназия - Интерактивные онлайн-курсы и тренажеры для развития мышления детей 6-13 лет SkillBox - это один из лидеров российского рынка онлайн-образования. Среди партнеров Skillbox ведущий разработчик сервисного дизайна AIC, медиа-компания Yoola, первое и самое крупное русскоязычное аналитическое агентство Tagline, онлайн-школа дизайна и иллюстрации Bang! Bang! Education, оператор PR-рынка PACO, студия рисования Draw&Go, агентство performance-маркетинга Ingate, scrum-студия Sibirix, имидж-лаборатория Персона. «Нетология» — это университет по подготовке и дополнительному обучению специалистов в области интернет-маркетинга, управления проектами и продуктами, дизайна, Data Science и разработки. В рамках Нетологии студенты получают ценные теоретические знания от лучших экспертов Рунета, выполняют практические задания на отработку полученных навыков, общаются с экспертами и единомышленниками. Познакомиться со всеми продуктами подробнее можно на сайте https://netology.ru, линейка курсов и профессий постоянно обновляется. StudyBay Brazil – это онлайн биржа для португалоговорящих студентов и авторов! Студент получает уникальную работу любого уровня сложности и больше свободного времени, в то время как у автора появляется дополнительный заработок и бесценный опыт. Автор24 — самая большая в России площадка по написанию учебных работ: контрольные и курсовые работы, дипломы, рефераты, решение задач, отчеты по практике, а так же любой другой вид работы. Сервис сотрудничает с более 70 000 авторов. Более 1 000 000 работ уже выполнено. StudyBay – это онлайн биржа для англоязычных студентов и авторов! Студент получает уникальную работу любого уровня сложности и больше свободного времени, в то время как у автора появляется дополнительный заработок и бесценный опыт.

Английская Википедия:Axiomatic foundations of topological spaces

Содержание

Standard definitions via open setsШаблон:Anchor

Definition via closed setsШаблон:Anchor

Definition via closure operatorsШаблон:Anchor

Definition via interior operatorsШаблон:Anchor

Definition via neighbourhoodsШаблон:Anchor

Definition via convergence of netsШаблон:Anchor

Definition via convergence of filtersШаблон:Anchor

See also

Citations

References

Навигация

Действия на странице

Действия на странице

Персональные инструменты

Навигация

Поиск

Инструменты