Sequential space

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 ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}$ 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 ${\displaystyle x_{j}\in I}$ (or said differently, if and only if for every neighborhood ${\displaystyle U}$ of ${\displaystyle x}$ and every ${\displaystyle i\in \mathbb {N} ,}$ ${\displaystyle U\cap x_{\geq i}\neq \varnothing }$).

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:

${\displaystyle {\begin{alignedat}{4}\operatorname {SeqCl} S:&=\left[S\right]_{\operatorname {seq} }:=\left\{x\in X~:~{\text{ there exists a sequence }}s_{\bullet }=\left(s_{i}\right)_{i=1}^{\infty }\subseteq S{\text{ in }}S{\text{ such that }}s_{\bullet }\to x{\text{ in }}(X,\tau )\right\}\end{alignedat}}}$

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:

${\displaystyle {\begin{alignedat}{4}\operatorname {SeqInt} S:&=\{s\in S~:~{\text{ whenever }}x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }\subseteq X{\text{ is a sequence in }}X{\text{ that converges to }}s{\text{ in }}(X,\tau ),{\text{ then }}x_{\bullet }{\text{ is eventually in }}S\}\\&=\{s\in S~:~{\text{ there does NOT exist a sequence }}x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }\subseteq X\setminus S{\text{ that converges in }}(X,\tau ){\text{ to a point in }}S\}\\&=X\setminus \operatorname {SeqCl} \left(X\setminus S\right)\end{alignedat}}}$

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

It is always true that ${\displaystyle \operatorname {SeqCl} \varnothing =\varnothing }$ and for all subsets ${\displaystyle R,S\subseteq X,}$

${\displaystyle \operatorname {SeqCl} _{X}\left(R\cup S\right)=\left(\operatorname {SeqCl} _{X}R\right)\cup \left(\operatorname {SeqCl} _{X}S\right).}$

For any ${\displaystyle S\subseteq X,}$

${\displaystyle S\subseteq \operatorname {SeqCl} _{X}S}$

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.

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 ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}$ 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 ${\displaystyle S}$ (i.e. there exists some integer ${\displaystyle i}$ such that the tail ${\displaystyle x_{\geq i}\subseteq S}$).
3. ${\displaystyle S=\operatorname {SeqOpen} _{X}S.}$
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 is 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.

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 ${\displaystyle x\in \left(\operatorname {Cl} S\right)\setminus S}$ 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 ${\displaystyle x\in \left(\operatorname {Cl} S\right)\setminus S,}$ 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 ${\displaystyle f:X\to Y}$ is called sequentially continuous if for every ${\displaystyle x\in X}$ and every sequence ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}$ 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 ${\displaystyle Y.}$
   • Every continuous map is necessarily sequentially continuous but in general, the conserve 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.

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 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]

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.

Every sequential space has countable tightness.

${\displaystyle \left(X,\operatorname {SeqOpen} (X,\tau )\right)}$ is a sequential space.[6]

The space ${\displaystyle \left(X,\operatorname {SeqOpen} (X,\tau )\right)}$ has the same convergent sequences and limits as ${\displaystyle (X,\tau ).}$ Explicitly, this means that if ${\displaystyle x\in X}$ and ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}$ is a sequence in ${\displaystyle X,}$ then ${\displaystyle \left(x_{i}\right)_{i=1}^{\infty }\to x}$ in ${\displaystyle (X,\tau )}$ if and only if ${\displaystyle \left(x_{i}\right)_{i=1}^{\infty }\to x}$ in ${\displaystyle \left(X,\operatorname {SeqOpen} (X,\tau )\right).}$

If ${\displaystyle \tau _{2}}$ is any topology on ${\displaystyle X}$ such that a sequence in ${\displaystyle X}$ converges to a point of ${\displaystyle X}$ in ${\displaystyle (X,\tau )}$ if and only if it does so in ${\displaystyle \left(X,\tau _{2}\right),}$ then necessarily ${\displaystyle \operatorname {SeqOpen} (X,\tau )=\operatorname {SeqOpen} \left(X,\tau _{2}\right).}$

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

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.

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".