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 formula_1 every open subset formula_2 (meaning that formula_3 and formula_4) has the following property: if a sequence formula_5 in formula_6 converges to some point in formula_2 then the sequence will be entirely contained in formula_2 (meaning that there exists an integer formula_9 such  that formula_10 all belong to formula_2); subset formula_3 with this property is said to be sequentially open, regardless of whether or not it is open in formula_13 In particular, every open subset is also sequentially open. However, it is possible for there to exist a subset formula_2 that has this property (of sequential openness) but is an open subset of formula_6 (that is, formula_16). Sequential spaces are 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 formula_6 where for any given  subset formula_18 knowledge of which sequences in formula_6 converge to which point(s) of formula_6 (and which sequences do not) is sufficient to determine whether or not formula_2 is closed in formula_22 In this way, sequential spaces are those spaces formula_6 for which sequences in formula_6 can be used as a "test" to correctly determine whether or not any given subset is open in formula_6 (specifically, this "test" is conducted by checking if the subset is sequentially open). In any space that is sequential, there exists a subset formula_2 for which this "test" gives a "false positive", meaning that formula_2 is  sequentially open (hence the "positive" test result) but it is not open (hence the "false" in "false positive"). Said differently, sequential spaces are those spaces whose topologies can be completely characterized in terms of sequence convergence. Alternatively, a space formula_28 being sequential means that its topology formula_29 if "forgotten", can be completely reconstructed using only sequences if one has possible information about the convergence/non-convergence of sequences in formula_28 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. There are other classes of topological spaces, such as Fréchet–Urysohn spaces, -sequential spaces, and formula_9-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_9-sequential space. Sequential spaces and formula_9-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_6 be a set and let formula_5 be a sequence in formula_36 where a is by  definition just a map from the natural numbers formula_37 into formula_22 If formula_2 is a set then formula_40 means that the formula_41 is a sequence in formula_42 If formula_43 is a map then formula_44 denotes the sequence formula_45 Because the sequence formula_41 is just a function formula_47 this is consistent with the definition of function composition, meaning that formula_48 For any index formula_49 the is the : formula_50 The set of all tails of formula_41 is denoted by formula_52 and it forms a (also called a ) on formula_36 which is why it is called the ' or the ' of formula_54 If formula_3  is a subset then a sequence formula_41 in formula_6 is if there exists some index formula_58 such that formula_59 (that is, formula_60 for any integer formula_61 such that formula_62). Let formula_28 be a topological space ( necessarily Hausdorff) and let formula_41 be a sequence in formula_22 The sequence formula_41 ' in formula_28 to a point formula_68 written formula_69 in formula_1 if for every neighborhood formula_71 of formula_72 in formula_1 formula_41 is eventually in formula_71; if this is the case then formula_72 is called a ' of formula_54 As usual, the notation formula_78 mean that formula_69 in formula_28 and formula_72 is the limit point  of formula_41 in formula_83 that is, if formula_84 in formula_28 then formula_86 If formula_28 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 ; every Hausdorff space is sequentially Hausdorff). A point formula_88 is called a ' or ' of formula_41 in formula_28 if for every neighborhood formula_71 of formula_72 in formula_28 and every formula_94 there exists some integer formula_62 such that formula_96 (or said differently, if and only if for every neighborhood formula_71 of formula_72 and every  formula_94 formula_100). Sequential closure/interior. Let formula_28 be a topological space and let formula_3 be a subset. The topological closure (resp. topological interior) of formula_2 in formula_28 is denoted by formula_105 (resp. formula_106). The of formula_2 in formula_28 is the set: formula_109 where formula_110 or formula_111 may be written if clarity is needed. The inclusion formula_112 always holds but in general, set equality may to hold. The is the map formula_113 defined by formula_114 where formula_115 denotes the power set of formula_22 The of formula_2 in formula_28 is the set: formula_119 where formula_120 or formula_121 may be written if clarity  is needed. As with the topological closure operator, formula_122 and formula_123 always hold and for all subsets formula_124 formula_125 so that consequently, formula_126 However, it is in general possible that formula_127 which in particular would imply that formula_128 because the topological closure operator is idempotent, meaning that formula_129 for all subsets formula_130 Transfinite sequential closure The is defined as follows: define formula_131 to be formula_132 define formula_133 to be formula_134 and for a limit ordinal formula_135 define formula_136 to be formula_137 Then there is a smallest ordinal formula_138 such that formula_139 and for this formula_135 formula_136 is called the transfinite  sequential closure of formula_142 In fact, formula_143 always holds where formula_144 is the first uncountable ordinal. The transfinite sequential closure of formula_145 is sequentially closed. Taking the transfinite sequential closure solves the idempotency problem above. The smallest formula_138 such that formula_147 for each formula_148 is called of the space formula_22 This ordinal invariant is well-defined for sequential spaces. Sequentially open/closed sets. Let formula_28 be a topological space ( necessarily Hausdorff) and let formula_3 be a subset. It is known that the subset formula_2 is open in formula_28 if and only if whenever formula_154 is a net in formula_6 that  converges in formula_28 to a point formula_157 then formula_41 is eventually in formula_159 where "" means that there exists some index formula_160 such that formula_60 for all formula_162 satisfying formula_163 The definition of a subset of formula_28 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_88 if it satisfies any of the following equivalent conditions: Let formula_169 denote the set of all sequentially open subsets of formula_1 where this may be denoted by formula_171 is the topology formula_172 is understood. Every open (resp. closed) subset of formula_6 is sequentially open (resp. sequentially closed), which implies that formula_174 It is possible for the containment formula_175 to be , meaning that there may exist a subset of formula_6 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_28  is called a if it satisfies any of the following equivalent conditions: By taking formula_178 and formula_179 to be the identity map on formula_6 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) if and only if (2): Assume that any sequentially open subsets is open and let formula_181 be sequentially closed. It is proved above that the complement formula_182 is sequentially open and thus open so that formula_181 is closed. The converse is similar. (2) if and only if (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_42 Since the limit is necessarily adherent to formula_159 it is in the closure of formula_42 Conversely, suppose for a contradiction that a subset formula_190 is sequentially closed but not closed. By 3, there exists a sequence in formula_181 that converges to a point in formula_192 that is, the limit lies outside formula_193 This contradicts sequential closedness of formula_193 formula_195 Section::::-sequential and formula_9-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_28 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_28 is called an ' (or ') if it satisfies any of the following equivalent conditions: Every first-countable space is formula_9-sequential. There exist topological vector spaces that  are sequential but formula_9-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_28 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 T space. Topology of sequentially open sets. Let formula_202 denote the set of all sequentially open subsets of the topological space formula_13 Then formula_202 is a topology on formula_6 that contains the original topology formula_206 that is, formula_207 Let formula_71 be sequentially open. It is now shown that its complement formula_209 is sequentially closed; that is, that a convergent sequence formula_5 of elements of formula_181 has its limit in formula_193 Suppose for a contradiction that formula_213 then there exists some integer formula_214 such that formula_215 which  contradicts the fact that all formula_216 are supposed to be in formula_193 The converse will now be shown; that is, it is now shown that if formula_181 is sequentially closed then its complement formula_182is sequentially open. Let formula_41 be a sequence in formula_6 such that formula_222 and suppose for a contradiction that for any formula_223 that is, for all integers formula_224 there exists formula_225 Define by recursion the subsequence formula_226 of elements of formula_181 : set formula_228 and then formula_229 that is, formula_230 and formula_231 It is convergent as a subsequence of a convergent sequence, and all its elements are in  formula_193 Hence the limit has to be in formula_233 which contradicts that formula_234 The sequence is therefore eventually in formula_235 It is now shown that the set of sequentially open subsets is a topology. Specifically, this means that formula_236 and formula_6 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_6 is eventually in formula_22 Let formula_240 be a family of sequentially open subsets, let formula_241 and let formula_41 be a sequence in formula_6 converging to formula_234  formula_72 being in the union means there exists formula_246 such that formula_247 and by sequential openness, the sequence is eventually in formula_248 Finally, if formula_249 is a finite intersection of sequentially open subsets, then a sequence converging to formula_250 eventually converges to each of the formula_251 that is, for all formula_252 satisfying formula_253 there exists some formula_254 such that formula_255 Taking formula_256 one has formula_257 The generated sequential topology is finer than the original one, meaning that if formula_71 is open, then it is sequentially open. Let formula_41 be a sequence in formula_6 converging to formula_234 Since formula_71 is open , it is a neighborhood of formula_72 and by definition of convergence, there exists formula_264 such that formula_265 formula_195 The topological space formula_28 is said to be if formula_202 is a Hausdorff space. Properties of the topology of sequentially open sets. Every sequential space has countable tightness. The topological space formula_269 is always a sequential space (even if formula_28 is not), and formula_269 has the same convergent sequences and limits as formula_13 Explicitly, this means that if formula_88 and formula_41 is a sequence in formula_36 then formula_69 in formula_28 if and only if formula_69 in formula_279 If formula_280 is any topology  on formula_6 such that for every formula_88 and every sequence formula_41 in formula_36 formula_285 then necessarily formula_286 If formula_287 is continuous then so is formula_288 Sequential continuity. A map formula_43 is said to be formula_88 if whenever formula_41 is a sequence in formula_6 such that formula_293 then formula_294 A map is said to be if it is sequentially continuous at every point of its domain. Equivalently, a map formula_287 sequentially continuous if and only if for every sequence formula_41 in formula_6 and every formula_68 if formula_69 in formula_28 then necessarily formula_301 in formula_302 which happens if and only if formula_303  is continuous. Every continuous map is sequentially continuous although in general, the converse may fail to hold. In fact, a space formula_28 is a sequential space if and only if it has the following : 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. Let formula_6 be a set and let formula_309 be a family of formula_6-valued maps with each map formula_311 being of the form formula_312 where the domain formula_313 is some topological space. If every domain formula_313 is a Fréchet–Urysohn space then the final topology on formula_6 induced by formula_309 makes formula_6 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 formula_318 and identify the set formula_319 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_320 denote the Schwartz space and let formula_321 denote the space of smooth functions on an open subset formula_322 where both of these spaces have their usual Fréchet space topologies , as defined in the article about distributions. Both formula_320 and formula_324 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_320 and formula_321 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. Spaces of test functions and distributions Let formula_327 denote the space of test  functions with its canonical LF topology, which makes it into a distinguished strict LF-space and let formula_328 denote the space of distributions, which by definition is the strong dual space of formula_329 These two spaces, 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_9-sequential spaces). Both formula_331 and formula_328 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 (that is, converges pointwise), which in particular, is the reason why a sequence of distributions converges in formula_328 (with is given strong dual topology) if and only if it converges pointwise. The space formula_331 is also a Schwartz topological vector space. Nevertheless, neither formula_331 nor its strong dual formula_328 is a sequential space (not even an Ascoli space). Cocountable topology 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_43 is a continuous open surjection between two Hausdorff sequential spaces then the set formula_338 is a closed subset of formula_36 the set formula_340 is a closed subset of formula_341 that satisfies formula_342 and the restriction formula_343 is injective. If formula_43 is a surjective map (not assumed to be continuous) onto a Hausdorff sequential  space formula_341 and if formula_346 is a basis for the topology on formula_36 then formula_43 is an open map if and only if for every formula_88 and every basic neighborhood formula_350 of formula_351 if formula_352 in formula_341 then necessarily formula_354 Here, formula_355 denotes the image (or range) of the sequence/map formula_356 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 (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".