순차공간

수학의 위상 및 관련 분야에서 순차 공간은 위상이 수렴/전달 순서에 의해 완전히 특성화될 수 있는 위상학적 공간이다. 그것들은 계산가능성에 대한 매우 약한 공리를 만족시키는 공간이라고 생각할 수 있다. 미터법 공간을 포함하는 모든 1차 카운트 가능 공간은 순차적 공간이다.

In any topological space ${\displaystyle (X,\tau ),}$ every open subset ${\displaystyle S}$ (meaning that ${\displaystyle S\subseteq X}$ and ${\displaystyle S\in \tau }$) has the following property: if a sequence ${\displaystyle x_{\bullet }=\left(x_{i}\rig$$ht)_{i=1}^{\infty }}$ in ${\displaystyle X}$ converges to some point in ${\displaystyle S}$ then the sequence will eventually be entirely contained in ${\displaystyle S}$ (meaning that there exists an integer ${\displaystyle N}$ such that ${\displaystyle x_{N},x_{N+1},\ldots }$ all be$길게$는 S ${\$$displaystyle$ $S},$ 이 속성을 가진 모든$$ 하위 집합 $S{\displaystyle }$ ${\displaystyle \subseteq }$ ${\displaystyle X}$ ${\displaystyle S\$$subseteq$ X$}$은$({\displaystyle X}$, $τ$ ${\displaystyle )}$에 열려 있는지 여부에 관계없이 순차적으로 열린다고 한다$.$$}}$ 특히$$ 모든 오픈 서브셋도 순차적으로 개방한다. 그러나 이러한 속성(순차 개방성)을 가지지만 X ${\displaystyle$ X$}$의 열린 부분 집합($$$즉{\displaystyle }$, S ${\displaystyle \notin }$ τ$$ {\$displaystyle S\not \in \tau$})$$이 아닌 부분$$ 집합 $S{\displaystyle }$ ${\displaystytle S}$이 존재할 수 있다. 순차적 공간은 정확히 위상학적 공간이며, 이 특성을 가진 부분집합은 절대 열리지 않는다. 닫힌 부분 집합의 관점에서 순차적 공간은 $X{\displaystyle }$ ${\displaystyle \{\backslash }$displaystyle ${\displaystyle X\},}$ ${\displaystyle S\subseteq X}$ 지식의$$ 경우 X ${\displaystyle$ $X$$}$의 시퀀스가 X ${\displaystyle$ X$}$의 어느 지점($$그리고 어떤 시퀀스가 충분하지 않은지)에 수렴하는$$ 공간$$ $X{\displaystyle }$ ${\displaystystysty$ X {\sty X}로 정확하게 볼 수 있다. $S{\displaystyle }$ ${\displaystyle$ S$}$이(가) X${\displaystyle }$. ${\displaystyle$ X$.}$에서 닫혔는지$$ 여부를 결정하십시오.

이러한 방식으로 순차적 $공간$은 $X{\displaystyle }$ ${\displaystyle X}$의 시퀀스를 "테스트"로 사용하여 $X{\displaystyle }$ ${\displaystyle X}$에 지정된 하위 집합이 열려 있는지 여부를 정확하게 판단할 수 있는$$$$ 공간 X ${\displaystyle X}$이다($$특히 이 "테스트"는 하위 집합이 순차적으로 열려 있는지 확인하여 수행됨). 순차적이지 않은 모든 공간에는 이 "테스트"가 "허위 양성"을 나타내는$$ 하위 $집합{\displaystyle }$ S ${\displaystyle$ S$}$이(가) 존재하며, $이$는 S ${\displaystyle S}$이(가) 순차적으로 열려 있지만("양성"으로 "허위 양성"으로 정의) 열리지 않는다는 것을 의미한다.^{[note 2]} 다르게 말하면, 순차 공간은 위상이 시퀀스 수렴 측면에서 완전히 특성화될 수 있는 공간이다.

또는 순차적 공간$$$({\displaystyle X}$, ${\displaystyle \tau }$) ${\displaystyle (X,\tau )}$은$({\displaystyle }$는) (X ${\displaystyle ,}$ ${\displaystyle \tau }$ ${\displaystyle )}$ {\displaystyle $\$tau ,}("잊어버린" 경우$$)의 위상 $\tau {\displaystyle }$, ${\$$displaystyle$ $\tau$ ,} 및 $(X,\tau)$에서 시퀀스의 수렴/비융합에 대한 가능한 모든 정보를 가진 경우에만 시퀀스를 사용하여 완전히 재구성할 수 있음을 의미한다. 다른 정보는 없다. 그러나, 모든 위상과 마찬가지로, 시퀀스 측면에서 완전히 설명할 수 없는 위상은 그럼에도 불구하고 그물(무어-스미스 시퀀스라고도 한다)의 관점에서 또는 필터의 측면에서 완전히 설명될 수 있다.

Frechet-Uryson 공간, T-sequential 공간 및 $N{\displaystyle }$ ${\displaystyle$ N$}-$sequential$$ 공간과 같은 다른 종류의 위상적 공간도 있는데, 이 공간은 공간의 위상이 시퀀스와 상호작용하는 방식 측면에서 정의되기도 한다. 이들의 정의는 순차 공간의 정의와 미묘하게(그러나 중요한) 방식으로만 다르며, 순차적 공간이 반드시 Fréchet-Uryson, T-sequential $또는{\displaystyle }$ N$${\$displaystyle N}-$sequential$$ space의 특성을 가지지 않는다는 것은 종종(초기) 놀라운 일이다.

순차 공간과 $N{\displaystyle }$ ${\displaystyle N}$ 순차$$ 공간은 S. P. 프랭클린이 도입했다.^[1]

역사

그러한 성질을 만족시키는 공간은 몇 년 동안 암묵적으로 연구되어 왔지만, 최초의 공식적 정의는 원래 1965년 S. P. 프랭클린에 기인하는데, 그는 "그들의 융합적 순서에 대한 지식으로 완전히 특정될 수 있는 위상학적 공간의 등급은 무엇인가?"라는 문제를 조사하던 중이었다. 프랭클린은 모든 1차 카운트 가능 공간은 그것의 수렴 순서에 대한 지식으로 완전히 지정될 수 있다는 점에 주목하여 위의 정의에 도달했고, 그리고 나서 그는 이것이 사실일 수 있도록 하는 1차 카운트 가능 공간의 특성을 추상화했다.

정의들

예선

Let ${\displaystyle X}$ be a set and let ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}$ be a sequence in ${\displaystyle X,}$ where a sequence in a set ${\displaystyle X}$ is by definition just a map from the natural numbers ${\displaystyle \mathbb {N}$$}$ into ${\displaystyle X.}$ If ${\displaystyle S}$ is a set then ${\displaystyle x_{\bullet }\subseteq S}$ means that the ${\displaystyle x_{\bullet }}$ is a sequence in ${\displaystyle S.}$ If ${\displaystyle f:X\to Y}$ is a map then ${\displaystyle f\left(x_{\bullet }\right):={(f(x_{}}_{}^{}}$${\$$=\left(f\left(x_{i}\right)\right)_{i=1}^{\infty }}$ denotes the sequence ${\displaystyle f\left(x_{1}\right),f\left(x_{2}\right),f\left(x_{3}\right),\ldots .}$ Because the sequence ${\displaystyle x_{\bullet }}$ is just a function ${\displaystyle$$x_{\bullet }:\mathb {N} X에 \}$ 이것은$$ 함수 구성의 정의와 일치하며$,$ 이는 $f{\displaystyle ({}_{}}$ ${\displaystyle {\bullet }_{}}$) ${\displaystyle :=}$ ${\displaystyle f(}$x ∙ ) ${\displaystyle {}_{}}$ ${\displaystyle {\bullet }_{}}$. ${\displaystyle f\left(x_{\bullet }\오른쪽$):$=f\ft x_{\ft }}$

모든 인덱스 $i{\displaystyle }$, ${\displaystyle$ i$}$에 대해 i ${\displaystyle$ i$}$에서 시작하는 $x{\displaystyle {}_{}}$ ${\displaystyle {\bullet }_{}}$ ${\$의 꼬리가 설정됨:

$x{\displaystyle {}_{}}$ ${\displaystyle {\bullet }_{}}$ ${\$의 모든 꼬리의 집합은 다음과 같이 표시된다$$.

그리고 $X{\displaystyle }$, ${\displaystyle X,}$에 필터 베이스(일명 프리필터라고도 함)를 형성하고 있어$$, 꼬리의 프리필터 또는 $x{\displaystyle {}_{}}$ ${\displaystyle {\bullet }_{}}$. {\$displaystyle x_{\bullet}}$의 꼬리의 순차 필터 베이스라고 불린다.

If ${\displaystyle S\subseteq X}$ is a subset then a sequence ${\displaystyle x_{\bullet }}$ in ${\displaystyle X}$ is eventually in ${\displaystyle S}$ if there exists some index ${\displaystyle i}$ such that ${\displaystyle x_{\geq i}\subseteq S}$ (that is, ${\$$j{\displaystyle }$ ${\displaystyle \ge }$ ${\displaystyle i}$ ${\displaystyle j}$과 같은$$ 정수 $j{\displaystyle }$ ${\$$displaystyle j}$에 $대해 displaystyle x_{j}\in$ S$}.$

Let ${\displaystyle (X,\tau )}$ be a topological space (not necessarily Hausdorff) and let ${\displaystyle x_{\bullet }}$ be a sequence in ${\displaystyle X.}$ The sequence ${\displaystyle x_{\bullet }}$ converges in ${\displaystyle (X,\tau )}$ to a point ${\display$$style x\in X,}$ written ${\displaystyle x_{\bullet }\to x}$ in ${\displaystyle (X,\tau ),}$ if for every neighborhood ${\displaystyle U}$ of ${\displaystyle x}$ in ${\displaystyle (X,\tau ),}$ ${\displaystyle x_{\bullet }}$ is eventually in ${\displaystyle U$$}$; if this is the case then ${\displaystyle x}$ is called a limit point of ${\displaystyle x_{\bullet }.}$ As usual, the notation ${\displaystyle \lim x_{\bullet }=x}$ mean that ${\displaystyle x_{\bullet }\to x}$ in ${\displaystyle (X,\tau )}$ and ${\displays$$tyle x}$ is the only limit point of ${\displaystyle x_{\bullet }}$ in ${\displaystyle (X,\tau );}$ that is, if ${\displaystyle x_{\bullet }\to z}$ in ${\displaystyle (X,\tau )}$ then ${\displaystyle z=a.}$ If ${\displaystyle (X,\tau )}$ is not Hausdorff 그러면 시퀀스가 둘 이상의 구별되는 지점(모든 시퀀스가 최대 한 지점까지 수렴되는 공간을 순차적인 하우스도르프 공간이라고 하며, 모든 하우스도르프 공간은 순차적으로 하우스도르프 공간이라고 한다.)으로 수렴하는 것이 가능할 수 있다.

A point ${\displaystyle x\in X}$ is called a cluster point or accumulation point of ${\displaystyle x_{\bullet }}$ in ${\displaystyle (X,\tau )}$ if for every neighborhood ${\displaystyle U}$ of ${\displaystyle x}$ in ${\displaystyle (X,\tau )}$ and every ${\di$$splaystyle i\in \mathbb {N} ,}$ there exists some integer ${\displaystyle j\geq i}$ such that ${\displaystyle x_{j}\in U}$ (or said differently, if and only if for every neighborhood ${\displaystyle U}$ of ${\displaystyle x}$ and every ${\displaystyle i\in \mathbb {N} ,}$ ${\displaystyle U\cap x_{\ge }}$${\displaystyle i_{}}$≠ ${\displaystyle \{}$ ${\displaystyle U\cap x_{\geq$ i$}\neq \varnothing }.$

순차 폐쇄/내부

(${\displaystyle X}$ ,${\displaystyle \tau }$) ${\displaystyle (X,\tau$ )}을$($를) 위상학적 $공간$으로 하고$$ S$\$ X {\$displaystyle$ S\$subseteq$X}을(를) 하위 집합으로$$ 한다. $({\displaystyle X}$, ${\displaystyle \tau }$) ${\displaystyle (X,\tau )}$에 $있는{\displaystyle }$ S ${\displaystyle S}$의 위상학적 폐쇄(resp. topological internal)는$$ ${\displaystyle {Cl}_{}}$ X ${\displaystyle {}_{}}$ ${\displaystyle S}$ ${\\displaystyle \operatorname {Cl} _{X}S}($resp)로 표시된다. ${\displaystyle {}_{}\mathrm{Int}}$ ${\displaystyle X_{}}$ ${\displaystyle {}_{}}$ S ${\displaystyle \operatorname {Int} _{X}S}$ $$.

$({\displaystyle X}$, ${\displaystyle \tau }$) ${\displaystyle (X,\tau )}$에서 $S{\displaystyle }$ ${\displaystyle S}$의 순차적인 폐쇄는 다음과$$ 같이 설정된다.

여기서 ${\displaystyle {\mathrm{SeqCl}}_{}}$ ${\displaystyle X_{}}$ ${\displaystyle {}_{}}$ ${\displaystyle S}$ ⁡ S $\displaystyle \operatorname {SeqCl} _{X}$${\displaystyle {\mathrm{SeqCl}}_{}}$( ${\displaystyle X_{}}$, ${\displaystyle t_{}}$ ) $)$ S$$${\displaysty \operatorname {SeqCl} _{(X,\tau$ )$S}$는 명확성이 필요할$$ 경우 작성할 수 있다. 포함된 ${\displaystyle {\mathrm{SeqCl}}_{}}$ ${\displaystyle X_{}}$ ${\displaystyle {}_{}}$ ${\displaystyle S}$ ${\displaystyle \subseteq }$ ${\displaystyle {\mathrm{Cl}}_{}}$ ${\displaystyle X_{}}$ ${\displaystyle {}_{}}$ ${\displaystyle S}$ ${\$은(는) 항상$$ 유지되지만 일반적으로 설정 균등이 유지되지 않을 수 있다. The sequential closure operator is the map ${\displaystyle \operatorname {SeqCl} _{X}:\wp (X)\to \wp (X)}$ defined by ${\displaystyle S\mapsto \operatorname {SeqCl} _{X}S,}$ where ${\displaystyle \wp (X)}$ denotes the power set of ${\displaystyle X.}$

$({\displaystyle X}$, ${\displaystyle \tau }$) ${\displaystyle (X,\tau )}$에서 $S{\displaystyle }$ ${\displaystyle S}$의 순차적 내부는 다음과 같이 설정된다$$.

여기서 ${\displaystyle {\mathrm{SeqInt}}_{}}$ ${\displaystyle X_{}}$ ${\displaystyle {}_{}}$ ${\displaystyle S}$ ${\displaystyle \operatorname {SeqInt} _$${$${\displaystyle {X\}}_{}}$$SeqInt$${\displaystyle {\}}_{}}$ $SeqStyle \operatorname {SeqInt} _{(X,\tau )S}$는 명확성이 필요할 경우 작성될 수 있다$$.

As with the topological closure operator, ${\displaystyle \,\operatorname {SeqCl} \varnothing =\varnothing \,}$ and ${\displaystyle \,\operatorname {SeqCl} X=X\,}$ always hold and for all subsets ${\displaystyle R,S\subseteq X,}$

그래서 결과적으로,

However, it is in general possible that ${\displaystyle \,\operatorname {SeqCl} _{X}\left(\operatorname {SeqCl} _{X}S\right)~\neq ~\operatorname {SeqCl} _{X}S\,}$ which in particular would imply that ${\displaystyle \,\operatorname {SeqCl} _{X}$$S\neq \operatorname {Cl} _{X}S\,}$ because the topological closure operator is idempotent, meaning that ${\displaystyle \,\operatorname {Cl} _{X}\left(\operatorname {Cl} _{X}S\right)$$~=~\operatorname {Cl} _{X}S\,$ 모든 하위 집합 $S{\displaystyle }$ ${\displaystyle \subseteq }$ ${\displaystyle X.}$ ${\displaystyle S\subseteq$ X$.}$에 대한$$ _{X}S\}

트랜스핀라이트 순차폐쇄

트랜스핀라이트 순차 폐쇄는 다음과 같이 정의된다: ${\displaystyle {A\mathrm{}}_{}}$ ${\displaystyle 0_{}}$ ${\$을 $A{\displaystyle ,}$ ${\displaystyle$ A}, $A$${\displaystyle {\mathrm{\alpha }}_{}}$+${\displaystyle {}_{}}$1 {\$displaystyle A_{\alpha +1}$을${\displaystyle {}_{}}$[${\displaystyle {{}_{}}_{}}$ ${\displaystyle {{\alpha }_{}}_{}}$ ${\displaystyle {{}_{}}_{}}$ ${\displaystyle {\mathrm{seq}}_{}}$, ${\displaystystyleftep]$로 정의한다$.$A_{\alpha}\right]_{\operatorname{seq}},}과 한계가 서수 α에,{\displaystyle \alpha,}가 되⋃β<>Aβ α.{\displaystyle \bigcup_{\beta<>\alpha}A_{\beta}.}그 때는 작은 서수 α{\displaystyle \alpha}그런 Aα한 α은 Aα{\displaystyle A_{\alpha}}을 정의 내린다.1+${\displaystyle A_{\alpha }=A_{\alpha +1},}$ and for this ${\displaystyle \alpha ,}$ ${\displaystyle A_{\alpha }}$ is called the transfinite sequential closure of ${\displaystyle A.}$ In fact, ${\displaystyle \alpha \leq \omega _{1},}$ always holds where ${\displaystyle \omega$$_{1$}은$($는) 최초의 헤아릴 수 없는 서수이다. $A{\displaystyle }$ ${\displaystyle A}$의 트랜스핀라이트 순차 폐쇄는 순차적으로 닫힌다$$. Transfinite 순차적 폐쇄를 취하면 위의 유휴 잠재력 문제가 해결된다. The smallest ${\displaystyle \alpha }$ such that ${\displaystyle A_{\alpha }={\overline {A}}}$ for each ${\displaystyle A\subseteq X}$ is called sequential order of the space ${\displaystyle X.}$^[2] This ordinal invariant is well-defined for sequential spaces.

순차적으로 열림/닫힘 세트

${\displaystyle \mathrm{Let}}$( ${\displaystyle X}$ ,${$ ) {\$displaystyle ($X,\$tau )}$은 위상학적 공간(필수적으로 하우스도르프는 아님)이고$$ $S{\displaystyle }$ ${\displaystyle \subseteq }$ ${\displaystyle X}$ ${\displaystyle S\subseteq$ X$}$은 부분 집합으로 한다$$. It is known that the subset ${\displaystyle S}$ is open in ${\displaystyle (X,\tau )}$ if and only if whenever ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ is a net in ${\displaystyle X}$ that converges in ${\displaystyle (X,\tau )}$ to a point $$${\displaystyle s\in S}$ then ${\displaystyle x_{\bullet }}$ is eventually in ${\displaystyle S,}$ where "eventually in ${\displaystyle S}$" means that there exists some index ${\displaystyle i\in I}$ such that ${\displaystyle x_{j}\in S}$ for all ${\displaystyle j\i$$n$ $만족{\displaystyle }$ j${\displaystyle i}$ i${\displaystyle .}${\$displaystyle$ j$\geq$ i$.}$$$ $({\displaystyle X}$, ${\displaystyle \tau }$) ${\displaystyle (X,\tau )}$의 순차적으로 열린 하위 집합의 정의는 망이 시퀀스로 대체되는 이 특성화의 변화를 사용한다$$.

$설정$된 S ${\displaystyle S}$이(가) 다음과 같은 동등한 조건 중 하나를 만족하는 경우 순차적으로 열린 상태로 호출된다$$.

1. 정의: $X{\displaystyle }$ ${\displaystyle X}$의 시퀀스가 $S{\displaystyle ,}$ ${\displaystyle$ S$,}$의 특정 지점으로 수렴될$$ 때마다 이 시퀀스는 결국 $S{\displaystyle .}$ ${\displaystyle$ S$.}$에 있게 된다$$.
2. If ${\displaystyle x_{\bullet }}$ is a sequence in ${\displaystyle X}$ and if there exists some ${\displaystyle s\in S}$ is such that ${\displaystyle x_{\bullet }\to s}$ in ${\displaystyle (X,\tau ),}$ then ${\displaystyle x_{\bullet }}$ is eventually in ${\d$$isplaystyle S}($즉, 꼬리 x $x{\displaystyle {}_{}}$ ${\displaystyle i_{}}$ ${\displaystyle {}_{}}$ ${\displaystyle s}$ ${\displaystyle x_{\geq i}\subseteq S})$과 같은$$ 정수 $i{\displaystyle }$ ${\displaystyle$ i$}$이(가) 있다.$$
3. ${\displaystyle S=\operatorname {SeqInt} _{X}S.}$
4. $세트$ X${\displaystyle s}$ S ${\displaystyle X\setminus S}$은$({\displaystyle }$ ${\displaystyle X}$ ,$τ$)에서 순차적으로 닫힌다$$$.$$}$

$설정$된 S ${\displaystyle S}$이(가) 다음과 같은 동등한 조건 중 하나를 만족하면 순차적으로 닫힘이라고 한다.

1. 정의: $S{\displaystyle }$ ${\displaystyle S}$의 시퀀스가${\displaystyle }$(${\displaystyle X}$, ${\displaystyle \tau }$) ${\displaystyle (X,\tau )}$에서 특정$$ 지점 ${\displaystyle x,}$ X$$, {\$displaystyle x$\$in$ X$,}$에 수렴될 때마다${\displaystyle x}$ ${\displaystyle .}$ S$$. {\$displaystyle$ x$\in$ S$.}$
2. If ${\displaystyle s_{\bullet }=\left(s_{i}\right)_{i=1}^{\infty }}$ is a sequence in ${\displaystyle S}$ and if there exists some ${\displaystyle x\in X}$ is such that ${\displaystyle s_{\bullet }\to x}$ in ${\displaystyle (X,\tau ),}$ then ${\displaystyle x\in }$${\displaystyle x\in$
3. ${\displaystyle S=\operatorname {SeqCl} S.}$
4. $세트{\displaystyle }$ X${\displaystyle s}$ S$${\$displaystyle X\setminus S}$은$({\displaystyle }$ ${\displaystyle X}$ ,$τ$)에서 순차적으로 열린다$$$.$$}$

순차적으로 오픈 세트의 보완은 순차적으로 닫힌 세트로서, 그 반대의 경우도 마찬가지다.

$세트{\displaystyle }$ S ${\displaystyle S}$이(가) ${\displaystyle \mathrm{다음}}$과 같은 동등한 조건 중 하나를 만족하는 경우$$ 점 $x{\displaystyle }$ ${\displaystyle \in }$ X ${\displaystyle x\in$ X$}$의 순차 근린이라고 한다.

1. $정의$: $x{\displaystyle }$ ${\displaystyle \in }$ ${\displaystyle \mathrm{SeqInt}}$ ${\displaystyle }$ ${\displaystyle S.}$ ${\displaystyle x\in \operatorname {SeqInt}$ S$.}$
   • ${\displaystyle \mathrm{중요}}$한 것은 "${\displaystyle }$ ${\displaystyle$ $S$$}$은(는) $x{\displaystyle }$ ${\displaystyle$ x$\}$의 순차적 인접 영역이며 "x ${\displaystyle \in }$ ${\displaystyle U.}$ ${\displaystyle x\in U\subseteq S$과($와{\displaystyle }$) 같은$$ 순차적으로 열린 집합 $U{\displaystyle }$ ${\displaystystyle U}$이(가 있음)로 정의되지 않는다.
2. $X{\displaystyle }$ ${\displaystyle x}$에 수렴되는$$ X ${\displaystyle$ X$}$의 $모든$ 시퀀스는 결국 S . $displaystyle$ S$.}$에 $있음$

내버려두다

$({\displaystyle X}$, ${\displaystyle \tau }$)${\displaystyle }$, ${\displaystyle (X,\tau })$의 모든 순차적으로 열려 있는 하위 집합 집합을 나타내며, 여기서$$ 이 하위 집합은 ${\displaystyle \mathrm{SeqOpen}}$ ${\displaystyle X}${\$displaysty \operatorname {SeqOpen} X$}이$$(가) $위상{\displaystyle }$ {$${\$displaysty \tau }$이(가 이해될 수 있다. $X{\displaystyle }$ ${\displaystyle X}$의 모든 열린(resp. closed) 부분 집합이$$ 순차적으로 열려 있음(resp. closed) 즉,

격납건물 $\tau {\displaystyle }$ ${\displaystyle \subseteq }$ ${\displaystyle \mathrm{SeqOpen}}$ ${\displaystyle (}$ (${\displaystyle X}$ ${\displaystyle ,}$ ${\displaystyle )}$ $){\displaystyle \tau \subseteq \operatorname {SeqOpen}(X,\tau$)}이(가) 적절할$$ 수 있으며, 이는 순차적으로 열리지만 열리지 않는$$ $X{\displaystyle }$ ${\displaysty X}$의 하위 집합이 존재할 수 있음을 의미한다. 마찬가지로, 닫히지 않은 순차적으로 닫힌 부분집합이 존재할 수 있다.

순차공간

위상학적 공간$({\displaystyle X}$ ,${\displaystyle )}$ ) ${\displaystyle (X,\tau )}$이(가) 다음과 같은 동등한 조건 중 하나를 만족하면 순차적 공간이라고 한다.

1. 정의: $X{\displaystyle }$ ${\displaystyle X}$의 모든 순차적으로 열린 하위 집합이 열려$$ 있다.
2. $X{\displaystyle }$ ${\displaystyle X}$의 모든 순차적으로 닫힌 부분 집합이 닫힌다$$.
3. {X\displaystyle,}에 그러한 조건이 S{S\displaystyle}에서 x에 전진 시퀀스 존재하는 일부 x ∈(당분이나 지방 말고도 X⁡ S)S∖{\displaystyle x\in \left(\operatorname{당분이나 지방 말고도}_{X}S\right)\setminus S} 살아 있을 모든 부분 집합 S⊆ X{\displaystyle S\subseteq X}에 대해, X에서,.{\displaystyle인데}는 경우에는 3은 닫히지 않았습니다.]
   • 이 조건을 프리셰-우린 공간의 다음과 같은 특성과 대조하십시오.

   For any subset ${\displaystyle S\subseteq X}$ that is not closed in ${\displaystyle X}$ and for every ${\displaystyle x\in \left(\operatorname {Cl} _{X}S\right)\setminus S,}$ there exists a sequence in ${\displaystyle S}$ that converges to ${\displaystyle x.}$

   • 이것은 모든 프래쳇-우르손 공간은 순차적인 공간임을 분명히 한다.
4. ${\displaystyle X}$ ${\displaystyle X}$은(는) 첫 번째 계산할 수 있는 공간의 몫이다.
5. ${\displaystyle X}$ ${\displaystyle X}$은(는) 메트릭 공간의 몫이다.
6. 순차적 공간의 범용 속성: 모든 위상학적 공간 $Y{\displaystyle }$, ${\displaystyle$ Y$,}$ 지도$$ $f{\displaystyle }$: ${\displaystyle X}$→ ${\displaystyle Y}$ ${\displaystyle f:X\to Y}$은 순차적으로 연속되는 경우에만 연속적이다.
   • A map ${\displaystyle f:(X,\tau )\to (Y,\sigma )}$ is called sequentially continuous if for every ${\displaystyle x\in X}$ and every sequence ${\displaystyle x_{\bullet }}$ in ${\displaystyle X,}$ if ${\displaystyle x_{\bullet }\to x}$ in ${\displaystyle X}$$$ then ${\displaystyle f\left(x_{\bullet }\right)=\left(f\left(x_{i}\right)\right)_{i=1}^{\infty }\to f(x)}$ in ${\displaystyle Y.}$ This condition is equivalent to the map ${\displaystyle f:(X,\opera$$토네이도 {SeqOpen}(X,\tau )\to(Y,\operatorname {SeqOpen}(Y,\sigma )}$은(는) 계속됨$$.
   • 모든 연속 지도는 반드시 순차적으로 연속되지만 일반적으로는 역이 유지되지 않을 수 있다.

마지막$$ $조건$에서 Y ${\displaystyle :=}$ ${\displaystyle X}$ ${\displaystyle$Y:=$X}$ 및 f$$ ${\displaystyle f}$을 $X{\displaystyle }$ ${\displaystyle X}$의 ID 맵으로$$ 삼음으로써 순차 공간의 클래스는 수렴 시퀀스에 의해 위상 구조가 결정되는 공간으로 정밀하게 구성된다.

T-sequential and ${\displaystyle N}$-sequential spaces

A sequential space may fail to be a T-sequential space and also a T-sequential space may fail to be a sequential space. In particular, it should not be assumed that a sequential space has the properties described in the next definitions.

A topological space ${\displaystyle (X,\tau )}$ is called a T-sequential space (or topological-sequential) if it satisfies one of the following equivalent conditions:^[1]

1. Definition: The sequential interior of every subset of ${\displaystyle X}$ is sequentially open.
2. The sequential closure of every subset of ${\displaystyle X}$ is sequentially closed.
3. For all ${\displaystyle S\subseteq X,}$${\displaystyle \operatorname {SeqCl} \left(\operatorname {SeqCl} S\right)=\operatorname {SeqCl} S.}$
   • The inclusion ${\displaystyle \operatorname {SeqCl} S\subseteq \operatorname {SeqCl} \left(\operatorname {SeqCl} S\right)}$ always holds for every ${\displaystyle S\subseteq X.}$
4. For all ${\displaystyle S\subseteq X,}$${\displaystyle \operatorname {SeqInt} S=\operatorname {SeqInt} \left(\operatorname {SeqInt} S\right).}$
   • The inclusion ${\displaystyle \operatorname {SeqInt} \left(\operatorname {SeqInt} S\right)\subseteq \operatorname {SeqInt} S}$ always holds for all ${\displaystyle S\subseteq X.}$
5. For all ${\displaystyle S\subseteq X,}$ ${\displaystyle \operatorname {SeqInt} S}$ is equal to the union of all subsets of ${\displaystyle S}$ that are sequentially open in ${\displaystyle (X,\tau ).}$
6. For all ${\displaystyle S\subseteq X,}$ ${\displaystyle \operatorname {SeqCl} S}$ is equal to the intersection of all subsets of ${\displaystyle X}$ that contain ${\displaystyle S}$ and are sequentially closed in ${\displaystyle (X,\tau ).}$
7. For all ${\displaystyle x\in X,}$ the collection of all sequentially open neighborhoods of ${\displaystyle x}$ in ${\displaystyle (X,\tau )}$ forms a neighborhood basis at ${\displaystyle x}$ for the set of all sequential neighborhoods of ${\displaystyle x.}$
   • This means for any ${\displaystyle x\in X}$ and any sequential neighborhood ${\displaystyle N}$ of ${\displaystyle x,}$ there exists a sequentially open set ${\displaystyle U}$ such that ${\displaystyle x\in U\subseteq N.}$
   • Here, the exact definition of "sequential neighborhood" is important because recall that "${\displaystyle N}$ is a sequential neighborhood of ${\displaystyle x}$" was defined to mean that ${\displaystyle x\in \operatorname {SeqInt} N.}$
8. For any ${\displaystyle x\in X}$ and any sequential neighborhood ${\displaystyle N}$ of ${\displaystyle x,}$ there exists a sequential neighborhood ${\displaystyle M}$ of ${\displaystyle x}$ such that for every ${\displaystyle m\in M,}$ the set ${\displaystyle N}$ is a sequential neighborhood of ${\displaystyle m.}$

As with T-sequential spaces, it should not be assumed that a sequential space has the properties described in the next definition.

A topological space ${\displaystyle (X,\tau )}$ is called an ${\displaystyle N}$-sequential (or neighborhood-sequential) space if it satisfies any of the following equivalent conditions:^[1]

1. Definition: For every ${\displaystyle x\in X,}$ if a set ${\displaystyle N\subseteq X}$ is a sequential neighborhood of ${\displaystyle x}$ then ${\displaystyle N}$ is a neighborhood of ${\displaystyle x}$ in ${\displaystyle (X,\tau ).}$
   • Recall that ${\displaystyle N}$ being a sequential neighborhood (resp. a neighborhood) of ${\displaystyle x}$ means that ${\displaystyle x\in \operatorname {SeqInt} N}$ (resp. ${\displaystyle x\in \operatorname {Int} N}$).
2. ${\displaystyle X}$ is both sequential and T-sequential.

Every first-countable space is ${\displaystyle N}$-sequential.^[1] There exist topological vector spaces that are sequential but not ${\displaystyle N}$-sequential (and thus not T-sequential).^[1] where recall that every metrizable space is first countable. There also exist topological vector spaces that are T-sequential but not sequential.^[1]

Fréchet–Urysohn spaces

Every Fréchet–Urysohn space is a sequential space but there exist sequential spaces that are not Fréchet–Urysohn.^[4][5] Consequently, it should not be assumed that a sequential space has the properties described in the next definition.

A topological space ${\displaystyle (X,\tau )}$ is called Fréchet–Urysohn space if it satisfies any of the following equivalent conditions:

1. Definition: For every subset ${\displaystyle S\subseteq X,}$ ${\displaystyle \operatorname {SeqCl} _{X}S=\operatorname {Cl} _{X}S.}$
2. Every topological subspace of ${\displaystyle X}$ is a sequential space.
3. For any subset ${\displaystyle S\subseteq X}$ that is not closed in ${\displaystyle X}$ and for every ${\displaystyle x\in \left(\operatorname {Cl} _{X}S\right)\setminus S,}$ there exists a sequence in ${\displaystyle S}$ that converges to ${\displaystyle x.}$

Fréchet–Urysohn spaces are also sometimes said to be Fréchet, which should not be confused with Fréchet spaces in functional analysis; confusingly, Fréchet space in topology is also sometimes used as a synonym for T₁ space.

Topology of sequentially open sets

Let ${\displaystyle \operatorname {SeqOpen} (X,\tau )}$ denote the set of all sequentially open subsets of the topological space ${\displaystyle (X,\tau ).}$ Then ${\displaystyle \operatorname {SeqOpen} (X,\tau )}$ is a topology on ${\displaystyle X}$ that contains the original topology ${\displaystyle \tau ;}$ that is, ${\displaystyle \tau \subseteq \operatorname {SeqOpen} (X,\tau ).}$

The topological space ${\displaystyle (X,\tau )}$ is said to be sequentially Hausdorff if ${\displaystyle \operatorname {SeqOpen} (X,\tau )}$ is a Hausdorff space.

Properties of the topology of sequentially open sets

Every sequential space has countable tightness.

The topological space ${\displaystyle (X,\operatorname {SeqOpen} (X,\tau ))}$ is always a sequential space (even if ${\displaystyle (X,\tau )}$ is not),^[6] and ${\displaystyle (X,\operatorname {SeqOpen} (X,\tau ))}$ has the same convergent sequences and limits as ${\displaystyle (X,\tau ).}$ Explicitly, this means that if ${\displaystyle x\in X}$ and ${\displaystyle x_{\bullet }}$ is a sequence in ${\displaystyle X,}$ then ${\displaystyle x_{\bullet }\to x}$ in ${\displaystyle (X,\tau )}$ if and only if ${\displaystyle x_{\bullet }\to x}$ in ${\displaystyle (X,\operatorname {SeqOpen} (X,\tau )).}$

If ${\displaystyle \sigma }$ is any topology on ${\displaystyle X}$ such that for every ${\displaystyle x\in X}$ and every sequence ${\displaystyle x_{\bullet }}$ in ${\displaystyle X,}$

then necessarily ${\displaystyle \operatorname {SeqOpen} (X,\tau )=\operatorname {SeqOpen} (X,\sigma ).}$

If ${\displaystyle f:(X,\tau )\to (Y,\sigma )}$ is continuous then so is ${\displaystyle f:(X,\operatorname {SeqOpen} (X,\tau ))\to (Y,\operatorname {SeqOpen} (Y,\sigma )).}$

Sequential continuity

A map ${\displaystyle f:X\to Y}$ is said to be sequentially continuous at a point ${\displaystyle x\in X}$ if whenever ${\displaystyle x_{\bullet }}$ is a sequence in ${\displaystyle X}$ such that ${\displaystyle x_{\bullet }\to x{\text{ in }}X,}$ then ${\displaystyle f\left(x_{\bullet }\right)\to f(x){\text{ in }}Y.}$

A map is said to be sequentially continuous if it is sequentially continuous at every point of its domain. Equivalently, a map ${\displaystyle f:(X,\tau )\to (Y,\sigma )}$ sequentially continuous if and only if for every sequence ${\displaystyle x_{\bullet }}$ in ${\displaystyle X}$ and every ${\displaystyle x\in X,}$ if ${\displaystyle x_{\bullet }\to x}$ in ${\displaystyle (X,\tau )}$ then necessarily ${\displaystyle f\left(x_{\bullet }\right)\to f(x)}$ in ${\displaystyle (Y,\sigma ),}$ which happens if and only if

is continuous.

Every continuous map is sequentially continuous although in general, the converse may fail to hold. In fact, a space ${\displaystyle (X,\tau )}$ is a sequential space if and only if it has the following universal property for sequential spaces:

For every topological space ${\displaystyle (Y,\sigma )}$ and every map ${\displaystyle f:X\to Y,}$ the map ${\displaystyle f:(X,\tau )\to (Y,\sigma )}$ is continuous if and only if it is sequentially continuous.

Sufficient conditions

Every first-countable space is sequential; thus every second-countable space, metric space, and discrete space is sequential. Every first-countable space is a Fréchet–Urysohn space and every Fréchet-Urysohn space is sequential. Thus every metrizable and pseudometrizable space is a sequential space and a Fréchet–Urysohn space.

A Hausdorff topological vector space is sequential if and only if there exists no strictly finer topology with the same convergent sequences.^[7][8]

Let ${\displaystyle X}$ be a set and let ${\displaystyle {\mathcal {F}}}$ be a family of ${\displaystyle X}$-valued maps with each map ${\displaystyle f\in {\mathcal {F}}}$ being of the form ${\displaystyle f:\left(Y_{f},\tau _{f}\right)\to X,}$ where the domain ${\displaystyle \left(Y_{f},\tau _{f}\right)}$ is some topological space. If every domain ${\displaystyle \left(Y_{f},\tau _{f}\right)}$ is a Fréchet–Urysohn space then the final topology on ${\displaystyle X}$ induced by ${\displaystyle {\mathcal {F}}}$ makes ${\displaystyle X}$ into a sequential space.

Examples

Every CW-complex is sequential, as it can be considered as a quotient of a metric space. The prime spectrum of a commutative Noetherian ring with the Zariski topology is sequential. Every premetric space is a sequential space.

Sequential spaces that are not first countable

Take the real line ${\displaystyle \mathbb {R} }$ and identify the set ${\displaystyle \mathbb {Z} }$ of integers to a point. It is a sequential space since it is a quotient of a metric space. But it is not first countable.

Sequential spaces that are not Fréchet–Urysohn spaces

The following extensively used spaces are prominent examples of sequential spaces that are not Fréchet–Urysohn spaces. Let ${\displaystyle {\mathcal {S}}\left(\mathbb {R} ^{n}\right)}$ denote the Schwartz space and let ${\displaystyle C^{\infty }(U)}$ denote the space of smooth functions on an open subset ${\displaystyle U\subseteq \mathbb {R} ^{n},}$ where both of these spaces have their usual Fréchet space topologies, as defined in the article about distributions. Both ${\displaystyle {\mathcal {S}}\left(\mathbb {R} ^{n}\right)}$ and ${\displaystyle C^{\infty }(U),}$ as well as the strong dual spaces of both these of spaces, are complete nuclear Montel ultrabornological spaces, which implies that all four of these locally convex spaces are also paracompact^[9] normal reflexive barrelled spaces. The strong dual spaces of both ${\displaystyle {\mathcal {S}}\left(\mathbb {R} ^{n}\right)}$ and ${\displaystyle C^{\infty }(U)}$ are sequential spaces but neither one of these duals is a Fréchet-Urysohn space.^[10][11]

Every infinite-dimensional Montel DF-space is a sequential space but not a Fréchet–Urysohn space.

Examples of non-sequential spaces

Spaces of test functions and distributions

Let ${\displaystyle C_{c}^{k}(U)}$ denote the space of test functions with its canonical LF topology, which makes it into a distinguished strict LF-space and let ${\displaystyle {\mathcal {D}}'(U)}$ denote the space of distributions, which by definition is the strong dual space of ${\displaystyle C_{c}^{\infty }(U).}$ These two spaces, which completely underpin the theory of distributions and which have many nice properties, are nevertheless prominent examples of spaces that are not sequential spaces (and thus neither Fréchet–Urysohn spaces nor ${\displaystyle N}$-sequential spaces).

Both ${\displaystyle C_{c}^{\infty }(U)}$ and ${\displaystyle {\mathcal {D}}'(U)}$ are complete nuclear Montel ultrabornological spaces, which implies that all four of these locally convex spaces are also paracompact^[9] normal reflexive barrelled spaces. It is known that in the dual space of any Montel space, a sequence of continuous linear functionals converges in the strong dual topology if and only if it converges in the weak* topology (that is, converges pointwise),^[12] which in particular, is the reason why a sequence of distributions converges in ${\displaystyle {\mathcal {D}}'(U)}$ (with is given strong dual topology) if and only if it converges pointwise. The space ${\displaystyle C_{c}^{\infty }(U)}$ is also a Schwartz topological vector space. Nevertheless, neither ${\displaystyle C_{c}^{\infty }(U)}$ nor its strong dual ${\displaystyle {\mathcal {D}}'(U)}$ is a sequential space (not even an Ascoli space).^[10][11]

Cocountable topology

Another example of a space that is not sequential is the cocountable topology on an uncountable set. Every convergent sequence in such a space is eventually constant, hence every set is sequentially open. But the cocountable topology is not discrete. In fact, one could say that the cocountable topology on an uncountable set is "sequentially discrete".

Properties

If ${\displaystyle f:X\to Y}$ is a continuous open surjection between two Hausdorff sequential spaces then the set ${\displaystyle R:=\left\{x\in X~:~f^{-1}(f(x))=\{x\}\,\right\}}$ is a closed subset of ${\displaystyle X,}$ the set ${\displaystyle S:=f(R)}$ is a closed subset of ${\displaystyle Y}$ that satisfies ${\displaystyle R=f^{-1}(S),}$ and the restriction ${\displaystyle f{\big \vert }_{R}:R\to Y}$ is injective.

If ${\displaystyle f:X\to Y}$ is a surjective map (not assumed to be continuous) onto a Hausdorff sequential space ${\displaystyle Y}$ and if ${\displaystyle {\mathcal {B}}}$ is a basis for the topology on ${\displaystyle X,}$ then ${\displaystyle f:X\to Y}$ is an open map if and only if for every ${\displaystyle x\in X}$ and every basic neighborhood ${\displaystyle B\in {\mathcal {B}}}$ of ${\displaystyle x,}$ if ${\displaystyle y_{\bullet }=\left(y_{i}\right)_{i=1}^{\infty }\to f(x)}$ in ${\displaystyle Y}$ then necessarily ${\displaystyle \varnothing \neq f(B)\cap \operatorname {Im} y_{\bullet }.}$ Here, ${\displaystyle \operatorname {Im} y_{\bullet }:=\left\{y_{i}:i\in \mathbb {N} \right\}}$ denotes the image (or range) of the sequence/map ${\displaystyle y_{\bullet }:\mathbb {N} \to Y.}$

Categorical properties

The full subcategory Seq of all sequential spaces is closed under the following operations in the category Top of topological spaces:

• Quotients
• Continuous closed or open images
• Sums
• Inductive limits
• Open and closed subspaces

The category Seq is not closed under the following operations in Top:

• Continuous images
• Subspaces
• Finite products

Since they are closed under topological sums and quotients, the sequential spaces form a coreflective subcategory of the category of topological spaces. In fact, they are the coreflective hull of metrizable spaces (that is, the smallest class of topological spaces closed under sums and quotients and containing the metrizable spaces).

The subcategory Seq is a Cartesian closed category with respect to its own product (not that of Top). The exponential objects are equipped with the (convergent sequence)-open topology. P.I. Booth and A. Tillotson have shown that Seq is the smallest Cartesian closed subcategory of Top containing the underlying topological spaces of all metric spaces, CW-complexes, and differentiable manifolds and that is closed under colimits, quotients, and other "certain reasonable identities" that Norman Steenrod described as "convenient".

See also

• Axioms of countability
• Closed graph – Graph of a map closed in the product space
• First-countable space – Topological space where each point has a countable neighbourhood basis
• Fréchet–Urysohn space
• Sequence covering map

Notes

Citations

References

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