| dbo:abstract |
En mathématiques, un espace séquentiel est un espace topologique dont la topologie est définie par l'ensemble de ses suites convergentes. C'est le cas en particulier pour tout espace à base dénombrable. (fr) 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 (especially metric spaces) are sequential. In any topological space if a convergent sequence is contained in a closed set then the limit of that sequence must be contained in as well. This property is known as sequential closure. Sequential spaces are precisely those topological spaces for which sequentially closed sets are in fact closed. (These definitions can also be rephrased in terms of sequentially open sets; see below.) Said differently, any topology can be described in terms of nets (also known as Moore–Smith sequences), but those sequences may be "too long" (indexed by too large an ordinal) to compress into a sequence. Sequential spaces are those topological spaces for which nets of countable length (i.e., sequences) suffice to describe the topology. Any topology can be refined (that is, made finer) to a sequential topology, called the sequential coreflection of The related concepts of Fréchet–Urysohn spaces, T-sequential spaces, and -sequential spaces are also defined in terms of how a space's topology interacts with sequences, but have subtly different properties. Sequential spaces and -sequential spaces were introduced by S. P. Franklin. (en) In topologia, uno spazio sequenziale è uno spazio topologico che soddisfa un assioma di numerabilità piuttosto debole. Gli spazi sequenziali costituiscono la più generale classe di spazi topologici per la quale le successioni di punti caratterizzano completamente la topologia. (it) 数学の位相空間論関連分野における列型空間（れつけいくうかん、れつがたくうかん、英: sequential space; 列状空間、列性空間）とは、開集合と閉集合が点列の収束で特長づけられる位相空間のことである。この空間上で定義された関数の連続性もまた、点列の収束性で特長づけられる。しかし列型空間であっても閉包の概念は点列の収束で特長づけられるとは限らず、これが可能な列型空間をフレシェ・ウリゾーン空間という。 位相空間が列型空間である必要十分条件はその空間が第一可算公理を満たす空間の商空間となることである。 空間にこうした可算性に関する条件が必要となるのは点列の概念がそもそも可算な全順序列として定義されているからであり、点列から可算性と全順序性の束縛を外した概念である有向点族の概念を用いれば空間に仮定を置くことなく収束で位相構造を特長づけられる。 任意の列型空間はを持つ。 (ja) 일반위상수학에서 점렬 공간(點列空間, 영어: sequential space)은 위상수학적 구조를 그물 대신 점렬만으로 다룰 수 있는 위상 공간이다. 점렬성은 제1 가산 공간의 조건을 매우 약화시킨 것이다. 점렬 공간의 범주는 범주론적으로 여러 좋은 성질들을 갖는다. (ko) In de topologie en aanverwante deelgebieden van de wiskunde, is een sequentiële ruimte een topologische ruimte, die voldoet aan een zeer zwak aftelbaarheidsaxioma. Sequentiële ruimten zijn de meest algemene klasse van ruimten, waarvoor rijen voldoende zijn om de topologie te bepalen. Elke sequentiële ruimte heeft een "telbare krapte". (nl) У топології секвенційним простором називається топологічний простір у якому властивість збіжності чи розбіжності послідовностей повністю визначає топологію. Поняття вперше формально ввів американський математик Стен Френклін у 1965 році. (uk) |
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https://arxiv.org/abs/math/0412558 http://matwbn.icm.edu.pl/ksiazki/fm/fm57/fm5717.pdf http://www.mathnet.ru/links/0411dc60fab54ffac1cb8172e57c8f69/rm5901.pdf%7Caccess-date=10 https://dergipark.org.tr/en/download/article-file/692156%7Caccess-date=10 http://projecteuclid.org/euclid.mmj/1028999711%7Caccess-date=10 http://projecteuclid.org/euclid.pjm/1102779712%7Caccess-date=10 http://matwbn.icm.edu.pl/ksiazki/fm/fm61/fm6115.pdf%7Caccess-date=10 |
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En mathématiques, un espace séquentiel est un espace topologique dont la topologie est définie par l'ensemble de ses suites convergentes. C'est le cas en particulier pour tout espace à base dénombrable. (fr) In topologia, uno spazio sequenziale è uno spazio topologico che soddisfa un assioma di numerabilità piuttosto debole. Gli spazi sequenziali costituiscono la più generale classe di spazi topologici per la quale le successioni di punti caratterizzano completamente la topologia. (it) 数学の位相空間論関連分野における列型空間（れつけいくうかん、れつがたくうかん、英: sequential space; 列状空間、列性空間）とは、開集合と閉集合が点列の収束で特長づけられる位相空間のことである。この空間上で定義された関数の連続性もまた、点列の収束性で特長づけられる。しかし列型空間であっても閉包の概念は点列の収束で特長づけられるとは限らず、これが可能な列型空間をフレシェ・ウリゾーン空間という。 位相空間が列型空間である必要十分条件はその空間が第一可算公理を満たす空間の商空間となることである。 空間にこうした可算性に関する条件が必要となるのは点列の概念がそもそも可算な全順序列として定義されているからであり、点列から可算性と全順序性の束縛を外した概念である有向点族の概念を用いれば空間に仮定を置くことなく収束で位相構造を特長づけられる。 任意の列型空間はを持つ。 (ja) 일반위상수학에서 점렬 공간(點列空間, 영어: sequential space)은 위상수학적 구조를 그물 대신 점렬만으로 다룰 수 있는 위상 공간이다. 점렬성은 제1 가산 공간의 조건을 매우 약화시킨 것이다. 점렬 공간의 범주는 범주론적으로 여러 좋은 성질들을 갖는다. (ko) In de topologie en aanverwante deelgebieden van de wiskunde, is een sequentiële ruimte een topologische ruimte, die voldoet aan een zeer zwak aftelbaarheidsaxioma. Sequentiële ruimten zijn de meest algemene klasse van ruimten, waarvoor rijen voldoende zijn om de topologie te bepalen. Elke sequentiële ruimte heeft een "telbare krapte". (nl) У топології секвенційним простором називається топологічний простір у якому властивість збіжності чи розбіжності послідовностей повністю визначає топологію. Поняття вперше формально ввів американський математик Стен Френклін у 1965 році. (uk) 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 (especially metric spaces) are sequential. Any topology can be refined (that is, made finer) to a sequential topology, called the sequential coreflection of Sequential spaces and -sequential spaces were introduced by S. P. Franklin. (en) |
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Espace séquentiel (fr) Spazio sequenziale (it) 列型空間 (ja) 점렬 공간 (ko) Sequentiële ruimte (nl) Sequential space (en) Секвенційний простір (uk) |
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