# Sequential space

Last updated

In topology and related fields of mathematics, a sequential space is a topological space that satisfies a very weak axiom of countability.

## Contents

In any topological space ${\displaystyle (X,\tau ),}$ every open subset ${\displaystyle S}$ has the following property: if a sequence ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}$ in ${\displaystyle X}$ converges to some point in ${\displaystyle S}$ then the sequence will eventually be entirely in ${\displaystyle S}$ (i.e. there exists an integer ${\displaystyle N}$ such that ${\displaystyle x_{N},x_{N+1},\ldots }$ all belong to ${\displaystyle S}$); Any set with this property is said to be sequentially open, regardless of whether or not it is open in ${\displaystyle (X,\tau ).}$ However, it is possible for there to exist a subset ${\displaystyle S}$ that has this property but fails to be an open subset of ${\displaystyle X.}$ Sequential spaces are exactly those topological spaces where a subset with this property never fails to be open. Sequential spaces can be viewed as exactly those spaces ${\displaystyle X}$ where for any single given subset ${\displaystyle S\subseteq X,}$ knowledge of which sequences in ${\displaystyle X}$ converge to which point(s) of ${\displaystyle X}$ (and which don't) is sufficient to determine whether or not ${\displaystyle S}$ is closed in ${\displaystyle X.}$ [note 1] Thus sequential spaces are those spaces ${\displaystyle X}$ for which sequences in ${\displaystyle X}$ can be used as a "test" to determine whether or not any given subset is open (or equivalently, closed) in ${\displaystyle X}$; or said differently, sequential spaces are those spaces whose topologies can be completely characterized in terms of sequence convergence. In any space that is not sequential, there exists a subset for which this "test" gives a "false positive." [note 2]

Alternatively, a space ${\displaystyle (X,\tau )}$ being sequential means that its topology ${\displaystyle \tau ,}$ if "forgotten", can be completely reconstructed using only sequences if one has all possible information about the convergence (or non-convergence) of sequences in ${\displaystyle (X,\tau )}$ and nothing more. However, like all topologies, any topology that cannot be described entirely in terms of sequences can nevertheless be described entirely in terms of nets (also known as Moore–Smith sequences) or alternatively, in terms of filters. All first-countable spaces, which includes metric spaces, are sequential spaces.

There are other classes of topological spaces, such as Fréchet–Urysohn spaces, T-sequential spaces, and ${\displaystyle N}$-sequential spaces, that are also defined in terms of how a space's topology interacts with sequences. Their definitions differ from that of sequential spaces in only subtle (but important) ways and it is often (initially) surprising that a sequential space does not necessarily have the properties of a Fréchet–Urysohn, T-sequential, or ${\displaystyle N}$-sequential space.

Sequential spaces and ${\displaystyle N}$-sequential spaces were introduced by S. P. Franklin. [1]

## History

Although spaces satisfying such properties had implicitly been studied for several years, the first formal definition is originally due to S. P. Franklin in 1965, who was investigating the question of "what are the classes of topological spaces that can be specified completely by the knowledge of their convergent sequences?" Franklin arrived at the definition above by noting that every first-countable space can be specified completely by the knowledge of its convergent sequences, and then he abstracted properties of first countable spaces that allowed this to be true.

## Definitions

### Preliminaries

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 ${\displaystyle f\left(x_{\bullet }\right):=\left(f\left(x_{i}\right)\right)_{i=1}^{\infty }}$ 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 ${\displaystyle x_{\bullet }:\mathbb {N} \to X,}$ this is consistent with the definition of function composition, meaning that ${\displaystyle f\left(x_{\bullet }\right):=f\circ x_{\bullet }.}$

For any index ${\displaystyle i,}$ the tail of ${\displaystyle x_{\bullet }}$ starting at ${\displaystyle i}$ is the set:

${\displaystyle x_{\geq i}:=\left\{x_{j}~:~i\leq j\in \mathbb {N} \right\}:=\left\{x_{i},x_{i+1},x_{i+2},\ldots \right\}.}$

The set of all tails of ${\displaystyle x_{\bullet }}$ is denoted by

${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right):=\left\{x_{\geq i}~:~i\in \mathbb {N} \right\}}$

and it forms a filter base (also called a prefilter) on ${\displaystyle X,}$ which is why it is called the prefilter of tails or the sequential filter base of tails of ${\displaystyle x_{\bullet }.}$

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, ${\displaystyle x_{j}\in S}$ for any integer ${\displaystyle j}$ such that ${\displaystyle j\geq i}$).

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 ${\displaystyle x\in X,}$ written ${\displaystyle x_{\bullet }\to x}$ in ${\displaystyle (X,\tau ),}$ and ${\displaystyle x}$ is called a limit point of ${\displaystyle x_{\bullet }}$ if for every neighborhood ${\displaystyle U}$ of ${\displaystyle x}$ in ${\displaystyle (X,\tau ),}$${\displaystyle x_{\bullet }}$ is eventually in ${\displaystyle U.}$ As usual, the notation ${\displaystyle \lim x_{\bullet }=x}$ mean that ${\displaystyle x_{\bullet }\to x}$ in ${\displaystyle (X,\tau )}$ and ${\displaystyle x}$ 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 then it is possible for a sequence to converge to two or more distinct points.

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 ${\displaystyle i\in \mathbb {N} ,}$ 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_{\geq i}\neq \varnothing }$).

### Sequential closure/interior

Let ${\displaystyle (X,\tau )}$ be a topological space and let ${\displaystyle S\subseteq X}$ be a subset. The topological closure (resp. topological interior) of ${\displaystyle S}$ in ${\displaystyle (X,\tau )}$ is denoted by ${\displaystyle \operatorname {Cl} _{X}S}$ (resp. ${\displaystyle \operatorname {Int} _{X}S}$).

The sequential closure of ${\displaystyle S}$ in ${\displaystyle (X,\tau )}$ is the set:

{\displaystyle {\begin{alignedat}{4}\operatorname {SeqCl} S:&=[S]_{\operatorname {seq} }:=\left\{x\in X~:~{\text{ there exists a sequence }}\,s_{\bullet }\subseteq S\,{\text{ in }}S{\text{ such that }}\,s_{\bullet }\to x{\text{ in }}(X,\tau )\right\}\end{alignedat}}}

where ${\displaystyle \operatorname {SeqCl} _{X}S}$ or ${\displaystyle \operatorname {SeqCl} _{(X,\tau )}S}$ may be written if clarity is needed. The inclusion ${\displaystyle \,\operatorname {SeqCl} _{X}S\subseteq \operatorname {Cl} _{X}S\,}$ always holds but in general, set equality may fail to hold. 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 ${\displaystyle X.}$

The sequential interior of ${\displaystyle S}$ in ${\displaystyle (X,\tau )}$ is the set:

{\displaystyle {\begin{alignedat}{4}\operatorname {SeqInt} S:&=\{s\in S~:~{\text{ whenever }}\,x_{\bullet }\subseteq X\,{\text{ is a sequence in }}X{\text{ that converges to }}s{\text{ in }}(X,\tau ),{\text{ then }}x_{\bullet }{\text{ is eventually in }}S\}\\&=\{s\in S~:~{\text{ there does NOT exist a sequence }}\,x_{\bullet }\subseteq X\setminus S\,{\text{ that converges in }}(X,\tau ){\text{ to a point in }}S\}\\&=X\,\setminus \,\operatorname {SeqCl} (X\setminus S)\end{alignedat}}}

where ${\displaystyle \operatorname {SeqInt} _{X}S}$ or ${\displaystyle \operatorname {SeqInt} _{(X,\tau )}S}$ may be written if clarity is needed.

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

${\displaystyle S~\subseteq ~\operatorname {SeqCl} _{X}S\qquad {\text{ and }}\qquad \operatorname {SeqCl} _{X}(R\cup S)~=~\left(\operatorname {SeqCl} _{X}R\right)~\cup ~\left(\operatorname {SeqCl} _{X}S\right)}$

so that consequently,

${\displaystyle \operatorname {SeqCl} _{X}S~\subseteq ~\operatorname {SeqCl} _{X}\left(\operatorname {SeqCl} _{X}S\right).}$

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 ${\displaystyle \,\operatorname {Cl} _{X}\left(\operatorname {Cl} _{X}S\right)~=~\operatorname {Cl} _{X}S\,}$ for all subsets ${\displaystyle S\subseteq X.}$

Transfinite sequential closure

The transfinite sequential closure is defined as follows: define ${\displaystyle A_{0}}$ to be ${\displaystyle A,}$ define ${\displaystyle A_{\alpha +1}}$ to be ${\displaystyle \left[A_{\alpha }\right]_{\operatorname {seq} },}$ and for a limit ordinal ${\displaystyle \alpha ,}$ define ${\displaystyle A_{\alpha }}$ to be ${\displaystyle \bigcup _{\beta <\alpha }A_{\beta }.}$ Then there is a smallest ordinal ${\displaystyle \alpha }$ such that ${\displaystyle A_{\alpha }=A_{\alpha +1},}$ 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 ${\displaystyle \omega _{1}}$ is the first uncountable ordinal. The transfinite sequential closure of ${\displaystyle A}$ is sequentially closed. Taking the transfinite sequential closure solves the idempotency problem above. 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.

### Sequentially open/closed sets

Let ${\displaystyle (X,\tau )}$ be a topological space (not necessarily Hausdorff) and let ${\displaystyle S\subseteq X}$ be a subset. 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 ${\displaystyle j\in I}$ satisfying ${\displaystyle j\geq i.}$ The definition of a sequentially open subset of ${\displaystyle (X,\tau )}$ uses a variation of this characterization in which nets are replaced with sequences.

The set ${\displaystyle S}$ is called sequentially open if it satisfies any of the following equivalent conditions:

1. Definition: Whenever a sequence in ${\displaystyle X}$ converges to some point of ${\displaystyle S,}$ then that sequence is eventually in ${\displaystyle S.}$
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 ${\displaystyle S}$ (that is, there exists some integer ${\displaystyle i}$ such that the tail ${\displaystyle x_{\geq i}\subseteq S}$).
3. ${\displaystyle S=\operatorname {SeqInt} _{X}S.}$
4. The set ${\displaystyle X\setminus S}$ is sequentially closed in ${\displaystyle (X,\tau ).}$

The set ${\displaystyle S}$ is called sequentially closed if it satisfies any of the following equivalent conditions:

1. Definition: Whenever a sequence in ${\displaystyle S}$ converges in ${\displaystyle (X,\tau )}$ to some point ${\displaystyle x\in X,}$ then ${\displaystyle x\in S.}$
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 ${\displaystyle x\in S.}$
3. ${\displaystyle S=\operatorname {SeqCl} S.}$
4. The set ${\displaystyle X\setminus S}$ is sequentially open in ${\displaystyle (X,\tau ).}$

The complement of a sequentially open set is a sequentially closed set, and vice versa.

The set ${\displaystyle S}$ is called a sequential neighborhood of a point ${\displaystyle x\in X}$ if it satisfies any of the following equivalent conditions:

1. Definition: ${\displaystyle x\in \operatorname {SeqInt} S.}$
• Importantly, "${\displaystyle S}$ is a sequential neighborhood of ${\displaystyle x}$" is not defined as: "there exists a sequentially open set ${\displaystyle U}$ such that ${\displaystyle x\in U\subseteq S.}$"
2. Any sequence in ${\displaystyle X}$ that converges to ${\displaystyle x}$ is eventually in ${\displaystyle S.}$

Let

{\displaystyle {\begin{alignedat}{4}\operatorname {SeqOpen} (X,\tau ):&=\left\{S\subseteq X~:~S{\text{ is sequentially open in }}(X,\tau )\right\}\\&=\left\{S\subseteq X~:~S=\operatorname {SeqInt} _{(X,\tau )}S\right\}\\\end{alignedat}}}

denote the set of all sequentially open subsets of ${\displaystyle (X,\tau ),}$ where this may be denoted by ${\displaystyle \operatorname {SeqOpen} X}$ is the topology ${\displaystyle \tau }$ is understood. Every open (resp. closed) subset of ${\displaystyle X}$ is sequentially open (resp. sequentially closed), which implies that

${\displaystyle \tau ~\subseteq ~\operatorname {SeqOpen} (X,\tau ).}$

It is possible for the containment ${\displaystyle \tau \subseteq \operatorname {SeqOpen} (X,\tau )}$ to be proper, meaning that there may exist a subset of ${\displaystyle X}$ that is sequentially open but not open. Similarly, it is possible for there to exist a sequentially closed subset that is not closed.

### Sequential spaces

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

1. Definition: Every sequentially open subset of ${\displaystyle X}$ is open.
2. Every sequentially closed subset of ${\displaystyle X}$ is closed.
3. For any subset ${\displaystyle S\subseteq X}$ that is not closed in ${\displaystyle X,}$there exists some ${\displaystyle x\in \left(\operatorname {Cl} _{X}S\right)\setminus S}$ for which there exists a sequence in ${\displaystyle S}$ that converges to ${\displaystyle x.}$ [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 ${\displaystyle x.}$
• This makes it obvious that every Fréchet–Urysohn space is a sequential space.
4. ${\displaystyle X}$ is the quotient of a first countable space.
5. ${\displaystyle X}$ is the quotient of a metric space.
6. Universal property of sequential spaces: For every topological space ${\displaystyle Y,}$ a map ${\displaystyle f:X\to Y}$ is continuous if and only if it is sequentially continuous.
• 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 ${\displaystyle f:(X,\operatorname {SeqOpen} (X,\tau ))\to (Y,\operatorname {SeqOpen} (Y,\sigma ))}$ being continuous.
• Every continuous map is necessarily sequentially continuous but in general, the converse may fail to hold.

By taking ${\displaystyle Y:=X}$ and ${\displaystyle f}$ to be the identity map on ${\displaystyle X}$ in the last condition, it follows that the class of sequential spaces consists precisely of those spaces whose topological structure is determined by convergent sequences.

Proof of the equivalences

(1) ${\displaystyle \iff }$ (2): Assume that any sequentially open subsets is open and let ${\displaystyle F}$ be sequentially closed. It is proved above that the complement ${\displaystyle U=X\setminus F}$ is sequentially open and thus open so that ${\displaystyle F}$ is closed. The converse is similar.

(2) ${\displaystyle \iff }$ (3): Contraposition of 2 says that "${\displaystyle S}$ not closed implies ${\displaystyle S}$ not sequentially closed", and hence there exists a sequence of elements of ${\displaystyle S}$ that converges to a point outside of ${\displaystyle S.}$ Since the limit is necessarily adherent to ${\displaystyle S,}$ it is in the closure of ${\displaystyle S.}$

Conversely, suppose for a contradiction that a subset ${\displaystyle S:=F}$ is sequentially closed but not closed. By 3, there exists a sequence in ${\displaystyle F}$ that converges to a point in ${\displaystyle {\overline {S}}\setminus S={\overline {F}}\setminus F,}$ i.e. the limit lies outside ${\displaystyle F.}$ This contradicts sequential closedness of ${\displaystyle F.}$${\displaystyle \blacksquare }$

### 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}=\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, ${\displaystyle \tau \subseteq \operatorname {SeqOpen} (X,\tau ).}$

Proofs

Let ${\displaystyle U}$ be sequentially open. It is now shown that its complement ${\displaystyle F=X\setminus U}$ is sequentially closed; that is, that a convergent sequence ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}$ of elements of ${\displaystyle F}$ has its limit in ${\displaystyle F.}$ Suppose for a contradiction that ${\displaystyle x_{n}\,{\underset {n\to \infty }{\longrightarrow }}\,x\in U,}$ then there exists some integer ${\displaystyle N>0}$ such that ${\displaystyle \left\lbrace x_{k}:k\geq N\right\rbrace \subset U,}$ which contradicts the fact that all ${\displaystyle x_{n}}$ are supposed to be in ${\displaystyle F.}$

The converse will now be shown; that is, it is now shown that if ${\displaystyle F}$ is sequentially closed then its complement ${\displaystyle U=X\setminus F}$is sequentially open. Let ${\displaystyle x_{\bullet }}$ be a sequence in ${\displaystyle X}$ such that ${\displaystyle \lim _{n\to \mathbb {N} }x_{n}=x\,\in U}$ and suppose for a contradiction that for any ${\displaystyle N\in \mathbb {N} ,\ \left\lbrace x_{k}:k\geq N\right\rbrace \not \subseteq U,}$ i.e. for all integers ${\displaystyle N>0,}$ there exists ${\displaystyle k_{N}\geq N,\ x_{k_{N}}\in F=X\setminus U.}$ Define by recursion the subsequence ${\displaystyle \left(x_{\varphi (n)}\right)_{n=1}^{\infty }}$ of elements of ${\displaystyle F}$ : set ${\displaystyle \varphi (0)=k_{0}}$ and then ${\displaystyle \varphi (n+1)=k_{\varphi (n)+1};}$ that is, ${\displaystyle x_{\varphi (0)}:=x_{k_{0}}}$ and ${\displaystyle x_{\varphi (n+1)}=x_{k_{\varphi (n)+1}}.}$ It is convergent as a subsequence of a convergent sequence, and all its elements are in ${\displaystyle F.}$ Hence the limit has to be in ${\displaystyle F,}$ which contradicts that ${\displaystyle x\in U.}$ The sequence is therefore eventually in ${\displaystyle U.}$

It is now shown that the set of sequentially open subsets is a topology. Specifically, this means that ${\displaystyle \varnothing }$ and ${\displaystyle X}$ are sequentially open, arbitrary unions of sequentially open subset is sequentially open and finite intersections of sequentially open subsets is sequentially open. Any empty sequence satisfies any property and any sequence in ${\displaystyle X}$ is eventually in ${\displaystyle X.}$ Let ${\displaystyle \left(U_{i}\right)_{i\in I}}$ be a family of sequentially open subsets, let ${\displaystyle U=\bigcup _{i\in I}U_{i},}$ and let ${\displaystyle x_{\bullet }}$ be a sequence in ${\displaystyle X}$ converging to ${\displaystyle x\in U.}$${\displaystyle x}$ being in the union means there exists ${\displaystyle i_{0}\in I}$ such that ${\displaystyle x\in U_{i_{0}}}$ and by sequential openness, the sequence is eventually in ${\displaystyle U_{i_{0}}.}$ Finally, if ${\displaystyle V=\bigcap _{i=1}^{n}U_{i}}$ is a finite intersection of sequentially open subsets, then a sequence converging to ${\displaystyle x\in V}$ eventually converges to each of the ${\displaystyle U_{i},}$ i.e. for all ${\displaystyle i\in \mathbb {N} }$ satisfying ${\displaystyle 1\leq i\leq n}$ there exists some ${\displaystyle N_{i}\in \mathbb {N} }$ such that ${\displaystyle \left\lbrace x_{k}:k\geq N_{i}\right\rbrace \subset U_{i}.}$ Taking ${\displaystyle N=\max _{1\leq i\leq n}N_{i},}$ one has ${\displaystyle \left\lbrace x_{k}:k\geq N\right\rbrace \subset V.}$

The generated sequential topology is finer than the original one, meaning that if ${\displaystyle U}$ is open, then it is sequentially open. Let ${\displaystyle x_{\bullet }}$ be a sequence in ${\displaystyle X}$ converging to ${\displaystyle x\in U.}$ Since ${\displaystyle U}$ is open, it is a neighborhood of ${\displaystyle x}$ and by definition of convergence, there exists ${\displaystyle N\in \mathbb {N} }$ such that ${\displaystyle \left\lbrace x_{k},\ k\geq N\right\rbrace \subset U.}$${\displaystyle \blacksquare }$

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 ${\displaystyle X,}$

${\displaystyle x_{\bullet }\to x{\text{ in }}(X,\tau )\qquad {\text{ if and only if }}\qquad x_{\bullet }\to x{\text{ in }}(X,\sigma ),}$

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 ${\displaystyle f:(X,\tau )\to (Y,\sigma )}$ is called sequentially continuous 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

${\displaystyle f:(X,\operatorname {SeqOpen} (X,\tau ))\to (Y,\operatorname {SeqOpen} (Y,\sigma ))}$

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, hence each 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.

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 space, 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 (i.e. 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 ${\displaystyle R:=\left\{x\in X~:~f^{-1}\left(f(x)\right)=\{x\}\,\right\}}$ 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:

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 (i.e., 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".

## Notes

1. This interpretation assumes that you make this determination only to the given set ${\displaystyle S}$ and not to other sets; said differently, you cannot simultaneously apply this "test" to infinitely many subsets (e.g. you can not use something akin to the axiom of choice). It is in Fréchet-Urysohn spaces that the closure of a set ${\displaystyle S}$ can be determined without it ever being necessary to consider any set other than ${\displaystyle S.}$ There exist sequential spaces that are not Fréchet-Urysohn spaces.
2. Although this "test" (which attempts to answer "is this set open (resp. closed)?") could potentially give a "false positive," it can never give a "false negative;" this is because every open (resp. closed) subset ${\displaystyle S}$ is necessarily sequentially open (resp. sequentially closed) so this "test" will never indicate "false" for any set ${\displaystyle S}$ that really is open (resp. closed).

## Citations

1. Snipes, Ray F. "T-sequential topological spaces"
• Arhangel'skiĭ, A. V.; Franklin, S. P. (1968). "Ordinal invariants for topological spaces". Michigan Math. J. 15 (3): 313–320. doi:.
2. Arkhangel'skii, A.V. and Pontryagin L.S.,  General Topology I, definition 9 p.12
3. Engelking 1989, Example 1.6.18
4. Ma, Dan (19 August 2010). "A note about the Arens' space" . Retrieved 1 August 2013.
5. Wilansky 2013, p. 224.
6. Dudley, R. M., On sequential convergence - Transactions of the American Mathematical Society Vol 112, 1964, pp. 483-507
7. "Topological vector space". Encyclopedia of Mathematics. Encyclopedia of Mathematics. Retrieved September 6, 2020. It is a Montel space, hence paracompact, and so normal.
8. Gabriyelyan, Saak "Topological properties of Strict LF-spaces and strong duals of Montel Strict LF-spaces" (2017)
9. T. Shirai, Sur les Topologies des Espaces de L. Schwartz, Proc. Japan Acad. 35 (1959), 31-36.
10. Trèves 2006, pp. 351-359.

## Related Research Articles

In mathematics, a continuous function is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to sufficiently small changes in its input. If not continuous, a function is said to be discontinuous. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, during which attempts such as the epsilon–delta definition were made to formalize it.

In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

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, open sets are a generalization of open intervals in the real line. In a metric space—that is, when a distance function is defined—open sets are the sets that, with every point P, contain all points that are sufficiently near to P.

In mathematics, the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S. The closure of S may equivalently be defined as the union of S and its boundary, and also as the intersection of all closed sets containing S. Intuitively, the closure can be thought of as all the points that are either in S or "near" S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior.

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, this implies that vector space operations be continuous functions. 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. This should not be confused with a closed manifold.

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 limit point of a set in a topological space is a point that can be "approximated" by points of in the sense that every neighbourhood of with respect to the topology on also contains a point of other than itself. A limit point of a set does not itself have to be an element of There is also a closely related concept for sequences. A cluster point or accumulation point of a sequence in a topological space is a point such that, for every neighbourhood of there are infinitely many natural numbers such that This definition of a cluster or accumulation point of a sequence generalizes to nets and filters. In contrast to sets, for a sequence, net, or filter, the term "limit point" is not synonymous with a "cluster/accumulation point"; by definition, the similarly named notion of a limit point of a filter refers to a point that the filter converges to.

In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function is open if for any open set in the image is open in Likewise, a closed map is a function that maps closed sets to closed sets. A map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa.

In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces are distinguished by that property that a linear map from a bornological space into any locally convex spaces is continuous if and only if it is a bounded linear operator.

In general topology and related areas of mathematics, the final topology on a set with respect to a family of functions from topological spaces into is the finest topology on that makes all those functions continuous.

In mathematics, a topological space is usually defined in terms of open sets. However, there are many equivalent characterizations of the category of topological spaces. Each of these definitions provides a new way of thinking about topological concepts, and many of these have led to further lines of inquiry and generalisation.

In the field of topology, a Fréchet–Urysohn space is a topological space with the property that for every subset the closure of in is identical to the sequential closure of in Fréchet–Urysohn spaces are a special type of sequential space.

In topology, a subfield of mathematics, filters are special families of subsets of a set that can be used to study topological spaces and define all basic topological notions such a convergence, continuity, compactness, and more. Filters also provide a common framework for defining various types of limits of functions such as limits from the left/right, to infinity, to a point or a set, and many others. Special types of filters called ultrafilters have many useful technical properties and they may often be used in place of arbitrary filters.

In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point towards which they all get closer to. The notion of "points that get progressively closer" is made rigorous by Cauchy nets or Cauchy filters, which are generalizations of Cauchy sequences, while "point towards which they all get closer to" means that this net or filter converges to Unlike the notion of completeness for metric spaces, which it generalizes, the notion of completeness for TVSs does not depend on any metric and is defined for all TVSs, including those that are not metrizable or Hausdorff.

In mathematics, an LB-space, also written (LB)-space, is a topological vector space that is a locally convex inductive limit of a countable inductive system of Banach spaces. This means that is a direct limit of a direct system in the category of locally convex topological vector spaces and each is a Banach space.

In functional analysis and related areas of mathematics, a metrizable topological vector space (TVS) is a TVS whose topology is induced by a metric. An LM-space is an inductive limit of a sequence of locally convex metrizable TVS.

In mathematics, particularly in functional analysis and topology, the closed graph theorem is a fundamental result stating that a linear operator with a closed graph will, under certain conditions, be continuous. The original result has been generalized many times so there are now many theorems referred to as "closed graph theorems."

In mathematics, specifically topology, a sequence covering map is any of a class of maps between topological spaces whose definitions all somehow relate sequences in the codomain with sequences in the domain. Examples include sequentially quotient maps, sequence coverings, 1-sequence coverings, and 2-sequence coverings. These classes of maps are closely related to sequential spaces. If the domain and/or codomain have certain additional topological properties then these definitions become equivalent to other well-known classes of maps, such as open maps or quotient maps, for example. In these situations, characterizations of such properties in terms of convergent sequences might provide benefits similar to those provided by, say for instance, the characterization of continuity in terms of sequential continuity or the characterization of compactness in terms of sequential compactness.