Sequential space

In topology and related fields of mathematics, a sequential space is a topological space that satisfies a very weak axiom of countability. Sequential spaces are the most general class of spaces for which sequences suffice to determine the topology.

Every sequential space has countable tightness.

Definitions

Let X be a topological space.

• A subset U of X is sequentially open if each sequence (xn) in X converging to a point of U is eventually in U (i.e. there exists N such that xn is in U for all n N.)
• A subset F of X is sequentially closed if, whenever (xn) is a sequence in F converging to x, then x must also be in F.

The complement of a sequentially open set is a sequentially closed set, and vice versa. Every open subset of X is sequentially open and every closed set is sequentially closed. The converses are not generally true. The sequentially open sets form a topology, which contains the original topology of the space and has the same notion of sequential convergence.

A sequential space is a space X satisfying one of the following equivalent conditions:

1. Every sequentially open subset of X is open.
2. Every sequentially closed subset of X is closed.
3. For any subset S  X that is not closed, i.e. such that ${\displaystyle {\overline {S}}\backslash S\neq \emptyset }$, there exists a sequence ${\displaystyle (x_{n})_{n\in \mathbb {N} }\in S^{\mathbb {N} }}$ of elements of S that converges to an element of ${\displaystyle {\overline {S}}\backslash S}$.[1]

That is, the original topology can be reconstructed by knowing the convergent sequences.

Sequential closure

Given a subset ${\displaystyle A\subseteq X}$ of a space ${\displaystyle X}$, the sequential closure ${\displaystyle [A]_{\text{seq}}}$ is the set

${\displaystyle [A]_{\text{seq}}=\{x\in X:\exists \{a_{n}\}\to x,a_{n}\in A\}}$

that is, the set of all points ${\displaystyle x\in X}$ for which there is a sequence in ${\displaystyle A}$ that converges to ${\displaystyle x}$. The map

${\displaystyle [\;]_{\text{seq}}:A\mapsto [A]_{\text{seq}}}$

is called the sequential closure operator. It shares some properties with ordinary closure, in that the empty set is sequentially closed:

${\displaystyle [\varnothing ]_{\text{seq}}=\varnothing .}$

Every closed set is sequentially closed:

${\displaystyle A\subseteq [A]_{\text{seq}}\subseteq {\overline {A}}}$

for all ${\displaystyle A\subseteq X}$; here ${\displaystyle {\overline {A}}}$ denotes the ordinary closure of the set ${\displaystyle A}$. Sequential closure commutes with union:

${\displaystyle [A\cup B]_{\text{seq}}=[A]_{\text{seq}}\cup [B]_{\text{seq}}}$

for all ${\displaystyle A,B\subseteq X}$. However, unlike ordinary closure, the sequential closure operator is not in general idempotent; that is, one may have that

${\displaystyle [A]_{\text{seq}}\subsetneq {\Big [}[A]_{\text{seq}}{\Big ]}_{\text{seq}},}$

and consequently ${\displaystyle [A]_{\text{seq}}\neq {\overline {A}}}$, even when ${\displaystyle A}$ is a subset of a sequential space ${\displaystyle X}$.

The transfinite sequential closure is defined as follows: define ${\displaystyle A_{0}}$ to be ${\displaystyle A}$, ${\displaystyle A_{\alpha +1}}$ to be ${\displaystyle [A_{\alpha }]_{\text{seq}}}$, and for a limit ordinal ${\displaystyle \alpha }$, ${\displaystyle A_{\alpha }}$ 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, we always have ${\displaystyle \alpha \leq \omega _{1}}$, where ${\displaystyle \omega _{1}}$ is the first uncountable ordinal.) 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 X.[2] This ordinal invariant is well-defined for sequential spaces.

Fréchet–Urysohn space

Topological spaces for which sequential closure is the same as ordinary closure are known as Fréchet–Urysohn spaces (such a space is also said to be Fréchet). That is, a Fréchet–Urysohn space has

${\displaystyle [A]_{\text{seq}}={\overline {A}}\,}$

for all ${\displaystyle A\subseteq X}$. A space is a Fréchet–Urysohn space if and only if every subspace is a sequential space. Every first-countable space is a Fréchet–Urysohn space.

The space is named after Maurice Fréchet and Pavel Urysohn.

Clearly, every Fréchet–Urysohn space is a sequential space. The opposite implication is not true in general.[3][4]

A topological space ${\displaystyle X}$ is a strong Fréchet–Urysohn space if for every point ${\displaystyle x\in X}$ and every sequence ${\displaystyle A_{1},A_{2},\ldots }$ of subsets of the space ${\displaystyle X}$ such that ${\displaystyle x\in \bigcap _{n}{\overline {A_{n}}}}$ , there are points ${\displaystyle x_{1}\in A_{1},x_{2}\in A_{2},\ldots }$ such that ${\displaystyle \{x_{n}\}\to x}$.

The above properties can be expressed as selection principles.

History

Although spaces satisfying such properties had implicitly been studied for several years, the first formal definition is originally due to S. P. Franklin (aka Stan 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.

Examples

Every first-countable space is sequential, hence each second-countable space, metric space, and discrete space is sequential. Further examples are furnished by applying the categorical properties listed below. For example, every CW-complex is sequential, as it can be considered as a quotient of a metric space.

There are sequential spaces that are not first countable. (One example is to take the real line R and identify the set Z of integers to a point.)

An 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".

Equivalent conditions

Many conditions have been shown to be equivalent to X being sequential. Here are a few:

• X is the quotient of a first countable space.
• X is the quotient of a metric space.
• For every topological space Y and every map f : X Y, we have that f is continuous if and only if for every sequence of points (xn) in X converging to x, we have (f(xn)) converging to f(x).

The final equivalent condition shows that the class of sequential spaces consists precisely of those spaces whose topological structure is determined by convergent sequences in the space.

Categorical properties

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

• 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 (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".