Sequential space

In topology and related fields of mathematics, a sequential space is a topological space that satisfies a very weak axiom of countability. 
In any topological space formula_1 every open subset formula_2 has the following property: if a sequence formula_3 in formula_4 converges to some point in formula_2 then the sequence will be entirely in formula_2 (i.e. there exists an integer formula_7 such that formula_8 all belong to formula_2); Any set with this property is said to be sequentially open, regardless of whether or not it is open in formula_10 
However, it is possible for there to exist a subset formula_2 that has this property but to be an open subset of formula_12 
Sequential spaces are those topological spaces where a subset with this property never fails to be open. 
Sequential spaces can be viewed as exactly those spaces formula_4 where for any single given subset formula_14 knowledge of which sequences in formula_4 converge to which point(s) of formula_4 (and which don't) is sufficient to determine whether or not formula_2 is closed in formula_12 
Thus sequential spaces are those spaces formula_4 for which sequences in formula_4 can be used as a "test" to determine whether or not any given subset is open (or equivalently, closed) in formula_4; or said differently, sequential spaces are those spaces whose topologies can be completely characterized in terms of sequence convergence. 
In any space that is sequential, there exists a subset for which this "test" gives a "false positive." 
Alternatively, a space formula_22 being sequential means that its topology formula_23 if "forgotten", can be completely reconstructed using only sequences if one has possible information about the convergence (or non-convergence) of sequences in formula_22 and . 
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, -sequential spaces, and formula_7-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 necessarily have the properties of a Fréchet–Urysohn, -sequential, or formula_7-sequential space. 
Sequential spaces and formula_7-sequential spaces were introduced by S. P. Franklin. 
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 formula_4 be a set and let formula_3 be a sequence in formula_30 where a is by definition just a map from the natural numbers formula_31 into formula_12 If formula_2 is a set then formula_34 means that the formula_35 is a sequence in formula_36 
If formula_37 is a map then formula_38 denotes the sequence formula_39 Because the sequence formula_35 is just a function formula_41 this is consistent with the definition of function composition, meaning that formula_42 
For any index formula_43 the is the :
The set of all tails of formula_35 is denoted by 
and it forms a (also called a ) on formula_30 which is why it is called the ' or the ' of formula_48 
If formula_49 is a subset then a sequence formula_35 in formula_4 is if there exists some index formula_52 such that formula_53 (that is, formula_54 for any integer formula_55 such that formula_56). 
Let formula_22 be a topological space ( necessarily Hausdorff) and let formula_3 be a sequence in formula_12 
The sequence formula_35 ' in formula_22 to a point formula_62 written formula_63 in formula_1 and formula_65 is called a ' of formula_35 if for every neighborhood formula_67 of formula_65 in formula_1 formula_35 is eventually in formula_71 
As usual, the notation formula_72 mean that formula_63 in formula_22 and formula_65 is the limit point of formula_35 in formula_77 that is, if formula_78 in formula_22 then formula_80 
If formula_22 is not Hausdorff then it is possible for a sequence to converge to two or more distinct points.
A point formula_82 is called a ' or ' of formula_35 in formula_22 if for every neighborhood formula_67 of formula_65 in formula_22 and every formula_88 there exists some integer formula_56 such that formula_90 (or said differently, if and only if for every neighborhood formula_67 of formula_65 and every formula_88 formula_94). 
Sequential closure/interior.
Let formula_22 be a topological space and let formula_49 be a subset. 
The topological closure (resp. topological interior) of formula_2 in formula_22 is denoted by formula_99 (resp. formula_100). 
The of formula_2 in formula_22 is the set:
where formula_104 or formula_105 may be written if clarity is needed. The inclusion formula_106 always holds but in general, set equality may to hold.
The is the map formula_107 defined by formula_108 where formula_109 denotes the power set of formula_12
The of formula_2 in formula_22 is the set:
where formula_114 or formula_115 may be written if clarity is needed. 
It is always true that formula_116 and for all subsets formula_117 
For any formula_14 
so that consequently, 
However, it is in general possible that formula_122 which in particular would imply that formula_123 because the topological closure operator is idempotent, meaning that formula_124 for all subsets formula_125
The is defined as follows: define formula_126 to be formula_127 define formula_128 to be formula_129 and for a limit ordinal formula_130 define formula_131 to be formula_132 
Then there is a smallest ordinal formula_133 such that formula_134 and for this formula_130 formula_131 is called the transfinite sequential closure of formula_137 
In fact, formula_138 always holds where formula_139 is the first uncountable ordinal. 
The transfinite sequential closure of formula_140 is sequentially closed. 
Taking the transfinite sequential closure solves the idempotency problem above. 
The smallest formula_133 such that formula_142 for each formula_143 is called of the space formula_12 This ordinal invariant is well-defined for sequential spaces.
Sequentially open/closed sets.
Let formula_22 be a topological space ( necessarily Hausdorff) and let formula_49 be a subset. It is known that the subset formula_2 is open in formula_22 if and only if whenever formula_149 is a net in formula_4 that converges in formula_22 to a point formula_152 then formula_35 is eventually in formula_154 where "" means that there exists some index formula_155 such that formula_54 for all formula_157 satisfying formula_158 The definition of a subset of formula_22 uses a variation of this characterization in which nets are replaced with sequences. 
The set formula_2 is called if it satisfies any of the following equivalent conditions:
The set formula_2 is called if it satisfies any of the following equivalent conditions:
The complement of a sequentially open set is a sequentially closed set, and vice versa.
The set formula_2 is called a of a point formula_82 if it satisfies any of the following equivalent conditions:
Let 
denote the set of all sequentially open subsets of formula_1 where this may be denoted by formula_166 is the topology formula_167 is understood. 
Every open (resp. closed) subset of formula_4 is sequentially open (resp. sequentially closed), which implies that 
It is possible for the containment formula_169 to be , meaning that there may exist a subset of formula_4 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 formula_22 is called a if it satisfies any of the following equivalent conditions:
By taking formula_173 and formula_174 to be the identity map on formula_4 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. 
(1) ⇔ (2): Assume that any sequentially open subsets is open and let formula_176 be sequentially closed. It is proved above that the complement formula_177 is sequentially open and thus open so that formula_176 is closed. The converse is similar. 
(2) ⇔ (3): Contraposition of 2 says that "formula_2 not closed implies formula_2 not sequentially closed", and hence there exists a sequence of elements of formula_2 that converges to a point outside of formula_36 Since the limit is necessarily adherent to formula_154 it is in the closure of formula_36 
Conversely, suppose for a contradiction that a subset formula_185 is sequentially closed but not closed. By 3, there exists a sequence in formula_176 that converges to a point in formula_187 i.e. the limit lies outside formula_188 This contradicts sequential closedness of formula_188 
-sequential and formula_7-sequential spaces.
A sequential space may fail to be a -sequential space and also a -sequential space may fail to be a sequential space. 
In particular, it should be assumed that a sequential space has the properties described in the next definitions. 
A topological space formula_22 is called a ' (or ') if it satisfies one of the following equivalent conditions:
As with -sequential spaces, it should be assumed that a sequential space has the properties described in the next definition. 
A topological space formula_22 is called an ' (or ') if it satisfies any of the following equivalent conditions:
Every first-countable space is formula_7-sequential. 
There exist topological vector spaces that are sequential but formula_7-sequential (and thus not -sequential). where recall that every metrizable space is first countable. 
There also exist topological vector spaces that are -sequential but not sequential.
Fréchet–Urysohn spaces.
Every Fréchet–Urysohn space is a sequential space but there exist sequential spaces that are not Fréchet–Urysohn. 
Consequently, it should be assumed that a sequential space has the properties described in the next definition. 
A topological space formula_22 is called if it satisfies any of the following equivalent conditions:
Fréchet–Urysohn spaces are also sometimes said to be , which should not be confused with Fréchet spaces in functional analysis; confusingly, in topology is also sometimes used as a synonym for T1 space.
Topology of sequentially open sets.
Let formula_196 denote the set of all sequentially open subsets of the topological space formula_10 
Then formula_196 is a topology on formula_4 that contains the original topology formula_200 that is, formula_201 
Let formula_67 be sequentially open. It is now shown that its complement formula_203 is sequentially closed; that is, that a convergent sequence formula_3 of elements of formula_176 has its limit in formula_188 
Suppose for a contradiction that formula_207 then there exists some integer formula_208 such that formula_209 which contradicts the fact that all formula_210 are supposed to be in formula_188
The converse will now be shown; that is, it is now shown that if formula_176 is sequentially closed then its complement formula_177is sequentially open.
Let formula_3 be a sequence in formula_4 such that formula_216 and suppose for a contradiction that for any formula_217 i.e. for all integers formula_218 there exists formula_219 Define by recursion the subsequence formula_220 of elements of formula_176 : set formula_222 and then formula_223 that is, formula_224 and formula_225 It is convergent as a subsequence of a convergent sequence, and all its elements are in formula_188 Hence the limit has to be in formula_227 which contradicts that formula_228 The sequence is therefore eventually in formula_71
It is now shown that the set of sequentially open subsets is a topology. Specifically, this means that formula_230 and formula_4 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 formula_4 is eventually in formula_12 Let formula_234 be a family of sequentially open subsets, let formula_235 and let formula_3 be a sequence in formula_4 converging to formula_228 formula_65 being in the union means there exists formula_240 such that formula_241 and by sequential openness, the sequence is eventually in formula_242 Finally, if formula_243 is a finite intersection of sequentially open subsets, then a sequence converging to formula_244 eventually converges to each of the formula_245 i.e. for all formula_246 satisfying formula_247 there exists some formula_248 such that formula_249 Taking formula_250 one has formula_251
The generated sequential topology is finer than the original one, meaning that if formula_67 is open, then it is sequentially open.
Let formula_3 be a sequence in formula_4 converging to formula_228 Since formula_67 is open, it is a neighborhood of formula_65 and by definition of convergence, there exists formula_258 such that formula_259
Properties of the topology of sequentially open sets.
Every sequential space has countable tightness. 
formula_260 is a sequential space. 
The space formula_260 has the same convergent sequences and limits as formula_10 Explicitly, this means that if formula_82 and formula_3 is a sequence in formula_30 then formula_266 in formula_22 if and only if formula_266 in formula_269 
If formula_270 is any topology on formula_4 such that a sequence in formula_4 converges to a point of formula_4 in formula_22 if and only if it does so in formula_275 then necessarily formula_276 
If formula_277 is continuous, then so is formula_278
A map formula_277 is called if for every sequence formula_280 in formula_4 and every formula_62 if formula_63 in formula_22 then necessarily formula_285 in formula_286 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 formula_22 is a sequential space if and only if it has the following : 
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.
Let formula_4 be a set and let formula_293 be a family of formula_4-valued maps with each map formula_295 being of the form formula_296 where the domain formula_297 is some topological space. If every domain formula_297 is a Fréchet–Urysohn space then the final topology on formula_4 induced by formula_293 makes formula_4 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.
Take the real line formula_302 and identify the set formula_303 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 formula_304 denote the Schwartz space and let formula_305 denote the space of smooth functions on an open subset formula_306 where both of these spaces have their usual Fréchet space topologies, as defined in the article about distributions. 
Both formula_304 and formula_308 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 normal reflexive barrelled spaces. The strong dual spaces of both formula_304 and formula_305 are sequential spaces but of these duals is a Fréchet-Urysohn space.
Every infinite-dimensional Montel DF-space is a sequential space but a Fréchet–Urysohn space.
Examples of non-sequential spaces.
Let formula_311 denote the space of test functions with its canonical LF topology, which makes it into a distinguished strict LF-space and let formula_312 denote the space of distributions, which by definition is the strong dual space of formula_313 
These two space, which completely underpin the theory of distributions and which have many nice properties, are nevertheless prominent examples of spaces that are sequential spaces (and thus neither Fréchet–Urysohn spaces nor formula_7-sequential spaces). 
Both formula_315 and formula_312 are complete nuclear Montel ultrabornological spaces, which implies that all four of these locally convex spaces are also paracompact normal reflexive barrelled spaces.
It is known that in the dual space of any Montel space, a of continuous linear functionals converges in the strong dual topology if and only if it converges in the weak* topology (i.e. pointwise), which in particular, is the reason why a sequence of distributions converges in formula_312 (with is given strong dual topology) if and only if it converges pointwise. 
The space formula_315 is also a Schwartz topological vector space. 
Nevertheless, neither formula_315 nor its strong dual formula_312 is a sequential space (not even an Ascoli space).
Another example of a space that is 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 formula_37 is a continuous open surjection between two Hausdorff sequential spaces then the set formula_322 is a closed subset of formula_30 the set formula_324 is a closed subset of formula_325 that satisfies formula_326 and the restriction formula_327 is injective.
If formula_37 is a surjective map (not assumed to be continuous) onto a Hausdorff sequential space formula_325 and if formula_330 is a basis for the topology on formula_30 then formula_37 is an open map if and only if for every formula_82 and every basic neighborhood formula_334 of formula_335 if formula_336 in formula_325 then necessarily formula_338 Here, formula_339 denotes the image (or range) of the sequence/map formula_340
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 closed under the following operations in Top:
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".