Sequential space

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. All first-countable spaces, which includes metric spaces, are sequential spaces.

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 ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}$ 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 belong to ${\displaystyle S}$); any subset ${\displaystyle S\subseteq X}$ with this property is said to be sequentially open, regardless of whether or not it is open in ${\displaystyle (X,\tau ).}$ In particular, every open subset is also sequentially open. However, it is possible for there to exist a subset ${\displaystyle S}$ that has this property (of sequential openness) but is not an open subset of ${\displaystyle X}$ (that is, ${\displaystyle S\not \in \tau }$). Sequential spaces are exactly those topological spaces where a subset with this property never fails to be open. In terms of closed subsets, sequential spaces can be viewed as exactly those spaces ${\displaystyle X}$ where for any given subset ${\displaystyle S\subseteq X,}$ knowledge of which sequences in ${\displaystyle X}$ converge to which point(s) of ${\displaystyle X}$ (and which sequences do not) is sufficient to determine whether or not ${\displaystyle S}$ is closed in ${\displaystyle X.}$^{[note 1]}

In this way, sequential spaces are those spaces ${\displaystyle X}$ for which sequences in ${\displaystyle X}$ can be used as a "test" to correctly determine whether or not any given subset is open in ${\displaystyle X}$ (specifically, this "test" is conducted by checking if the subset is sequentially open). In any space that is not sequential, there exists a subset ${\displaystyle S}$ for which this "test" gives a "false positive", meaning that ${\displaystyle S}$ is sequentially open (hence the "positive" test result) but it is not open (hence the "false" in "false positive").^{[note 2]} Said differently, sequential spaces are those spaces whose topologies can be completely characterized in terms of sequence convergence.

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/non-convergence of sequences in ${\displaystyle (X,\tau )}$ and no other information. 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.

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:

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

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 ${\displaystyle (X,\tau ),}$ 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 ${\displaystyle x}$ 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 might be possible for a sequence to converge to two or more distinct points (a space in which every sequence converges to at most one point is called a sequentially Hausdorff space; every Hausdorff space is sequentially Hausdorff).

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:

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:

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 ${\displaystyle R,S\subseteq X,}$

so that consequently,

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

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

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]
   • Contrast this condition to the following characterization of a Fréchet–Urysohn space:

   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.

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 ).^[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[]

• Arkhangel'skii, A.V. and Pontryagin, L.S., General Topology I, Springer-Verlag, New York (1990) ISBN 3-540-18178-4.
• Arkhangel'skii, A V (1966). "Mappings and spaces" (PDF). Russian Mathematical Surveys. 21 (4): 115–162. Bibcode:1966RuMaS..21..115A. doi:10.1070/RM1966v021n04ABEH004169. ISSN 0036-0279. Retrieved 10 February 2021.
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