History
Although spaces satisfying such properties had implicitly been studied for several years, the first formal definition is due to S. P. Franklin in 1965. Franklin wanted to determine "the classes of topological spaces that can be specified completely by the knowledge of their convergent sequences", and began by investigating the first-countable spaces, for which it was already known that sequences sufficed. Franklin then arrived at the modern definition by abstracting the necessary properties of first-countable spaces.
Preliminary definitions
See also: Filters in topology and Net (mathematics)
Let be a set and let be a sequence in ; that is, a family of elements of , indexed by the natural numbers. In this article, means that each element in the sequence is an element of and, if is a map, then For any index the tail of starting at is the sequence
A sequence is eventually in if some tail of satisfies
Let be a topology on and a sequence therein. The sequence converges to a point written (when context allows, ), if, for every neighborhood of eventually is in is then called a limit point of
A function between topological spaces is sequentially continuous if implies
Sequential closure/interior
Let be a topological space and let be a subset. The topological closure (resp. topological interior) of in is denoted by (resp. ).
The sequential closure of in is the set
which defines a map, the sequential closure operator, on the power set of If necessary for clarity, this set may also be written or It is always the case that but the reverse may fail.
The sequential interior of in is the set
(the topological space again indicated with a subscript if necessary).
Sequential closure and interior satisfy many of the nice properties of topological closure and interior: for all subsets
- and ;
Proof
Fix If then there exists with But by the definition of sequential interior, eventually is in contradicting
Conversely, suppose ; then there exists a sequence with that is not eventually in By passing to the subsequence of elements not in we may assume that But then
▮
- and ;
- ;
- ; and
That is, sequential closure is a preclosure operator. Unlike topological closure, sequential closure is not idempotent: the last containment may be strict. Thus sequential closure is not a (Kuratowski) closure operator.
Sequentially closed and open sets
A set is sequentially closed if ; equivalently, for all and such that we must have note 1
A set is defined to be sequentially open if its complement is sequentially closed. Equivalent conditions include:
- or
- For all and such that eventually is in (that is, there exists some integer such that the tail ).
Sequential spaces and coreflection
As discussed above, sequential closure is not in general idempotent, and so not the closure operator of a topology. One can obtain an idempotent sequential closure via transfinite iteration: for a successor ordinal define (as usual)
and, for a limit ordinal define
This process gives an ordinal-indexed increasing sequence of sets; as it turns out, that sequence always stabilizes by index (the first uncountable ordinal). Conversely, the sequential order of is the minimal ordinal at which, for any choice of the above sequence will stabilize.2
The transfinite sequential closure of is the terminal set in the above sequence: The operator is idempotent and thus a closure operator. In particular, it defines a topology, the sequential coreflection. In the sequential coreflection, every sequentially-closed set is closed (and every sequentially-open set is open).3
Sequential spaces
A topological space is sequential if it satisfies any of the following equivalent conditions:
- is its own sequential coreflection.4
- Every sequentially open subset of is open.
- Every sequentially closed subset of is closed.
- For any subset that is not closed in there exists somenote 2 and a sequence in that converges to 5
- (Universal Property) For every topological space a map is if and only if it is (if then ).
T- and N-sequential spaces
A T-sequential space is a topological space with sequential order 1, which is equivalent to any of the following conditions:1
- The sequential closure (or interior) of every subset of is sequentially closed (resp. open).
- or are idempotent.
- or
- Any sequential neighborhood of can be shrunk to a sequentially-open set that contains ; formally, sequentially-open neighborhoods are a neighborhood basis for the sequential neighborhoods.
- For any and any sequential neighborhood of there exists a sequential neighborhood of such that, for every the set is a sequential neighborhood of
Being a T-sequential space is incomparable with being a sequential space; there are sequential spaces that are not T-sequential and vice-versa. However, a topological space is called a -sequential (or neighborhood-sequential) if it is both sequential and T-sequential. An equivalent condition is that every sequential neighborhood contains an open (classical) neighborhood.1
Every first-countable space (and thus every metrizable space) is -sequential. There exist topological vector spaces that are sequential but not -sequential (and thus not T-sequential).1
Fréchet–Urysohn spaces
Main article: Fréchet–Urysohn space
A topological space is called Fréchet–Urysohn if it satisfies any of the following equivalent conditions:
Consequences
Every sequential space has countable tightness and is compactly generated.
If is a continuous open surjection between two Hausdorff sequential spaces then the set of points with unique preimage is closed. (By continuity, so is its preimage in the set of all points on which is injective.)
If is a surjective map (not necessarily continuous) onto a Hausdorff sequential space and bases for the topology on then is an open map if and only if, for every basic neighborhood of and sequence in there is a subsequence of that is eventually in
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 (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 , , and and that is closed under colimits, quotients, and other "certain reasonable identities" that described as "convenient".
Notes
- note 1You cannot simultaneously apply this "test" to infinitely many subsets (for example, you can not use something akin to the axiom of choice). Not all sequential spaces are Fréchet-Urysohn, but only in those spaces can the closure of a set can be determined without it ever being necessary to consider any set other than
- note 2A Fréchet–Urysohn space is defined by the analogous condition for all (not "some") such : For any subset that is not closed in for any there exists a sequence in that converges to
