Sequences in topological spaces*

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I: PRACE MATEMATYCZNE X I (1968)

a n n at.e s s o c ie t a t i s m a t h e m a t i c a e p o l o n a e Series I: COMMENTATIONES MATHEMATICAE X I (1968)

С. E .

(Blacksburg, Virginia)

Sequences in topological spaces*

Recently there has been a renewed interest in sequences in con­

nection w ith «^-classes of Frechet p) and w ith topological spaces and their interrelations. See for instance K isy ń sk i [1 1 ], Dudley [4] and Fra n klin [5] (2).

We w ill be particularly interested in topological spaces (X,&~) with either of the following properties.

(a) I f M is a subset that contains the sequential lim its of all convergent sequences, then M is closed.

(b) I f x e X , M <= X such that x e M ' , then there exists a sequences {Xnf, x n e M such that x n converges to x.

I t is known that (b) (a). Topological spaces satisfying (b) and such that sequences converge to at most one point (espace 8 of Fróchet) have been studied by Frechet [6] and Urysohn [13]. Topological spaces satisfying (a) and their relations to topological spaces satisfying (b) have been studied by Hausdorff [9], K isy ń sk i [11], Dudley [4] and Fra n klin [5].

The concept of a side point of a sequence plays an important role in th is paper. (A side point is an accumulation point of the set of values of a sequence such that no subsequence of the sequence converges to the point.) I t w ill be proved that in topological spaces satisfying the condition that sequences converge to at most one point that each of the following conditions follow from the previous ones; (b), (a), every countably compact subset is closed, every sequentially compact subset is closed, convergent sequences lack side points. I f compact subsets are closed, the last con­

dition is satisfied. Locally sequentially compact spaces that satisfy the condition that sequentially compact subsets are closed sa tisfy (a). Fu rth e r -

* Most of the research was done at Kent State University. Most of the results were announced in the American Mathematical Society Notices (1965).

P) For a discussion of basic properties of ^-classes, see Kuratowski [12].

(2) Recently at the second Prague Topology Symposium a paper on this subject was given by V. Koutnik relating to work of J. Novak.

more in locally sequentially compact spaces, highly divergent sequences (sequences without convergent subsequences) lack side points.

Definitions of various types compactness are the same as in Kelley [10], i.e., a topological space is sequentially compact if every sequence has a convergent subsequence (the sequential lim it is in the space).

A topological space is countably compact if every countable open cover has a fin ite subcover. In a 1\ space th is is equivalent to every infinite subset having an accumulation point. A topological space (X,&~) is locally sequentially compact (countably) compact if for every x e X , there is a sequentially (countably) compact neighborhood of x.

Side points

De f i n i t i o n

1. A point

is a side po in t of a sequence

if у is an accumulation point of the set of values of {жп} but no subsequence of {xn} converges to y.

In a T x space th is is equivalent to saying that the sequence {xn}

is frequently in every neighborhood of у but not eventually in every neighborhood of y.

De f i n i t i o n

2. A sequence is highly divergent if it has no convergent subsequence.

Th e o r e m

1. A topological space ( X, S T) is sequentially compact i ff it has no highly divergent sequence. A T x space is countably compact i f f every highly divergent sequence has a side point.

P ro o f. The f irs t statement follows from D efinition 2. Let (X , 2 T ) be countably compact and let {xn} be highly divergent. {xn} takes on any value a finite number of times. Let M = [#„]. There is a point y e 31', since ( X , У ) is countably compact; у is then a side point of {;v On the other hand, let every highly divergent sequence have a side point.

Let 31 be an in fin ite set and P a countably in fin ite subset of M. I f among the sequences of distinct points formed from points of P there is a con­

vergent sequence, then P has an accumulation point. I f a ll sequences are highly divergent, then 31 has an accumulation point in th is case. So ( X, &~) is countably compact.

Th e o r e m

2. I n a T x space {X, 2T) every countably compact subset is sequentially compact i f every sequence has a subsequence without a side point.

P ro o f. Let 31 be countably compact. I t w ill be sufficient to consider the case when M is infinite . Le t {xn} be a sequence of distinct points of 31. {xn} has a subsequence {yn} without side points. Set N = U [yn] ; N must have a lim it point у e 31 and since у is not a side point of {y n}, {yn}

must have a convergent subsequence. I t follows that any sequence of

points of M w ill have a convergent subsequence.

Sequences in topological spaces 3 3 1

Th e o r e m 3.

I n locally sequentially compact spaces no highly divergent sequence has a side point.

P ro o f. Let у be a side point of a highly divergent sequence {xn}.

Let N y be a sequentially compact neighborhood of y. There is a subsequence of {xn} in N y taking on distinct values which must have a convergent subsequence which contradicts that {xn} is highly divergent.

The referee has pointed out that as a consequence of Theorems 1 and 3 locally sequentially compact, countably compact T x spaces are sequentially compact.

Axioms related to sequences

De f i n i t i o n

3. A topological space (X,&~) satisfies S 0: I f sequences converge to at most one point.

I f S 0 is satisfied and every convergent (highly divergent) sequence has a subsequence without side points (3).

S 2(H 2): I f S 0 is satisfied and no convergent (highly divergent) sequence has a side point.

S 3(S4): I f every sequentially (countably) compact subset is closed.

S 5: I f S 0 is satisfied and every sequentially closed set is closed. A set is sequentially closed if it contains the sequential lim its of a ll convergent sequences.

S6: I f S 0 is satisfied and for x e X , M a X such that x e M' , there is a sequence {x^} of points of M converging to x.

Z : I f S 0 is satisfied and for x e X , M a X such that xe M' , there is a subset P с M such that [x ] = P '.

The next theorem relates these axioms. An ELS,- space is a topological space satisfying both Hi and A ->■ В means that a topological spaces satisfying axiom A satisfies axiom B .

Th e o r e m 4.

(a) H 1S 3 - > S 4, (b) S 5 H a-> H 4, (c) S n+1-^ S W, (d) S 6 -* Z -> S4, (e) Z H x, (f) Z S 5 = S 6.

P ro o f, (a) Follow s from Theorem 2, (b) H 2 -> H 4 is immediate.

Let {jxn} be highly divergent. Let M = (J | ]хп]. M ~ [ y i y e X ] is se­

quentially closed and hence closed. So S 5 - > H 2. (c) S 2 S i -> S 0 is immediate from D efinition 3. Let {xn} be a convergent sequence, con­

verging to a point x. Set M — U [xn]. \xn~\ w M r**j [У‘У Ф x] is se­

quentially compact. So S 3 -> S 2. Since a sequentially compact subset in a T x space is countably compact S 4 -> S 3. S 5 -> S 3 since a sequentially compact set in an S„ space is sequentially closed by (a) and (b) S 5 -> S 4.

S 6 -> S 5 is a known result, (d) S 6 Z is immediate. Le t M be countably compact. Let x e M' . I f Z is satisfied, there is a subset P с M, such that

(3) For S0 spaces H^Sx is equivalent to the statement that every sequence has a subsequence without side points.

[ж] — Р' . Since М is countably compact, х е 31. So Z - > S 4. (e) Let {xn}

be highly divergent and let w be a side point of {xn}. Since Z is satisfied there is a subsequence {yn} of which w is the only side point; since {yn}

does not converge to w, there is a subsequence of { yn} without any side points. Hence Z H x. (f) Let ( X , ZT) be Z S 5. Let же 31'. There is P a 31 such that [ж] = P '. I f P ~ [ж] is sequentially closed, by the S 5 property P r'-' [#] is closed. Hence there is a sequence in P converging to ж. So S 6 is satisfied. B y (c) and (d) Z S 5 = S 6.

Clearly in Z H 2 spaces, no sequence has a side point. A countable space is S 6 if f no sequence has a side point. B y Theorem 3 locally sequen­

tia lly compact topological spaces sa tisfy H 2.

Th e o r e m

5. Locally sequentially (countably) compact S 3(Z) spaces satisfy S 5(S6). Locally countably compact H x spaces satisfy H 2.

P ro o f. Let 31 be sequentially closed. Le t же 31'. Let N x be a sequen­

tia lly compact neighborhood of x. N x ^ M is sequentially compact and by the S 3 property it is closed; so х е Жх ^ M and M is closed. The firs t statement follows. How let (X , be locally countably compact and sa tisfy Z . Since Z -^ Н -^ , by Theorem 2, ( X , ^ ~ ) is locally sequentially compact. Le t же 31'. There is a subset P c i such that [ж] — P '. Let N x be sequentially compact. N x гл P is sequentially compact and infinite w ith ж as the sole lim it point; hence there is a sequence of distinct points of P and hence of M converging to ж. S 6 is then satisfied. Le t a locally countably compact space sa tisfy H x. Le t w be a side point of a highly divergent sequence {xn}. There is a countably compact neighborhood N w containing a subsequence of distinct points of {xn}, {yn}. {yn} has a subsequence of distinct points without any side points contradicting the countable compactness of N w. So H 2 is satisfied.

I t might be said that local sequential compactness has the property of improving the sequential behavior of topological spaces; local-countably compact spaces tend to behave either very poorly or reasonably well in regard to sequences. As we w ill see later the Stone-Cech compactification of the integers does not even sa tisfy H x.

Properties of ^{S 0-S2} spaces. In a previous paper, the author (1) proved that an S 0 space is metrizable if f it has a (7-locally finite base and is coun­

tably paracompact. T h is latter condition may be replaced by local countable paracompactness; see A u ll [2]. Analogous to a theorem about T 2 spaces we have the following theorem about countable filte rs. See Gaal

[7, 261].

Th e o r e m

6. A topological space is S 0 i f f every countable filter has at

most one lim it point.

Sequences in topological spaces 333

Proo f. B y constructing the Fróchet filte r for a given convergent sequence, the necessity is established. Let {Bn} be a countable base for a convergent filte r converging to a set A containing at least 2 distinct points x and y. Eve ry N x гл N y contains B n for some n. Construct a sequence {xn} such that x n eB n . {xn} w ill converge to x and y, contrary to the S 0 property.

We now tu rn to intersection properties of sequentially compact and countably compact subsets.

Th e o r e m

7. I n S 0 spaces the intersection of sequentially compact subsets is sequentially compact. I n spaces the intersection o f a sequentially compact and a countably compact subset is sequentially compact.

Proo f. Let M = П M a where each M a is sequentially compact. Let {xn} be a sequence such that x n e M . {x n} has a convergent subsequence {yn} converging to y. B y the S 0 property у

M a for each a. So у

M.

Le t Ж be sequentially compact and N countably compact in an S x space. Let {xn} be such that xn e M r\ N . {xn} has a convergent subse­

quence {yn} converging to у and y e M . {yn} has a convergent subsequence without a side point. Since N is countably compact, y e N .

Clearly in or S 4 spaces the intersection of an arbitrary fam ily of countably compact subsets is countably compact.

The S 2 spaces include a ll T 2 spaces and there are T 2 spaces that do not sa tisfy any of the other axioms discussed here.

Th e o r e m

8. A topological space such that any compact set is closed is an S 2 space.

P ro o f. H . Cullen [3,123] has proved that such spaces are S 0. Let {Xn\ be a sequence convergent to x. Le t Ж = U [a?n] and let у Ф x.

(Ж

[ж]) ^ [y] is compact and hence closed so {xn} has no side points.

Note th is same argument can be used to show that (X is S 0.

I t follows that T 2 spaces sa tisfy S 2 since in T 2 spaces compact subsets are closed (4).

Further remarks on classification. The F-spaces discussed in Gillman and Jerison [8] are a class of spaces w ith no in fin ite convergent sequence and hence sa tisfy S 8. Some like (JN do not satisfy S 4.

The E spaces discussed by the author [2], spaces such that every point is the intersection of countable neighborhoods is a large class of spaces including countable T 2 spaces and perfectly normal spaces which sa tisfy S4 but not necessarily S 5 or Z .

Fo r a discussion of classes of spaces satisfying S 5 and S 6, see Dudley [4] and Fra n klin [5].

(4) For some further relations involving S„ and the condition that any compact set is closed see Wilansky [14].

I t is interesting to note that if a given topology satisfies S 0, S 4, S 2, S 3, or S 4 any finer topologies satisfy these axioms.

Examples. A series of examples are given to show the independence of the axioms. Some of the properties, particularly the compactness properties of several of the examples are well known results.

The following example due to S. Fra n klin [5], p. 110 and 113, satisfies S 5 but not S 6.

Ex a m p l e

1. Let X be the real numbers w ith the topology generated by the usual topology and a ll sets of the form [0] w V where V is a usual open neighborhood of the sequence {1/ri}.

T h is example also illu stra te s that a space may satisfy S 5 and hence have no side points for either convergent or highly divergent sequences and yet have a sequence w ith side points. In th is example 0 is a side point of a sequence of a ll the relations excluding points of the form 1 jn.

Ex a m p l e

2. Let (X be the uncountable product of closed unit intervals w ith the usual topology. Since (X,&~) is T 2, it satisfies S x by Theorems 4 and 8. B u t ( X , 3~) is known to be countably compact without being sequentially compact; so by Theorem 2, {X, 3~) does not satisfy H x.

Ex a m p l e

3. Let X be the real line and if f ~ T is countable or if T = 0 . T h is example satisfies S 4 and H 2 but not Z or S 5.

P ro o f. (X , J7") satisfies S 4 since a ll countably compact subsets are finite. I t satisfies H 2 since a ll countable subsets are closed. (X,&~) does not sa tisfy Z since every point of the space is an accumulation point of every uncountable subset and countable subsets have no accumulation points. S 5 is not satisfied as a ll proper uncountable subsets are sequen­

tia lly closed but not closed.

The next example is due to Arens. See Kelley [10], p. 77.

Ex a m p l e

4. Le t X be the set of a ll pairs of non-negative integers w ith the topology described as follows: Fo r each point ( m, n) other than (0, 0) the set {(m, n)} is open. A set U is a neighborhood of (0, 0) if f for a ll except a fin ite number of integers m the set {n: (m, ri)4~U} is finite.

T h is example satisfies Z and consequently and S 4 but it does not sa tisfy S 5 or H 2.

P ro o f. [0, 0] is the only lim it point in ( X , ^ ~ ) so Z is satisfied; yet [0, 0] is not the sequential lim it of any convergent sequence so H 2 is not satisfied and by Theorem 4, S 5 is not satisfied. [0, 0] is a side point of any sequence w ith range consisting of a ll points of the space.

Ex a m p l e

5. Let X be an uncountable discrete space. Add a new

point c giving a new space X c. Le t the topology °U for X c consist of a ll

sets open in X and the complements of closed and countable sets of X .

Sequences in topological spaces 3 3 5

(Essentia lly the Lindelóf analogy to the one point compactification).

( X c, %) satisfies Z and H 2 but not S 5.

Pro o f. { Х С, Щ satisfies Z since there is only one lim it point in the space and it satisfies H 2 since a ll countable sets are closed. I t does not satisfy S 5 since X is sequentially closed but not closed in X c.

E

6. The Stone-Cech compactification of j3N of the positive integers is S 3 but not S4 or H 1.

Proof. Every sequentially compact subset is finite. See Gillman and Jerison [8], p. 208 and 215. There is a countably compact subset in fiN which is not compact and not closed so (3N is not S 4. See Gillman and Jerison [8], p. 135.

The author is in debt to S. Fra n klin for pointing out some interesting properties of the above example. The next example due to S. Fra n klin is an example of an S 3 space w ith a compact subset that is not closed.

E

7. Modify ftN by duplicating one point x of (3N — N and keep the same neighborhoods except both points are closed. Let у be the new point. The complement of the positive integers and у is compact but not closed.

In contrast to the above example, the next example shows a topolo­

gical space in which compact subsets are closed but not a ll sequentially compact subsets are closed.

E

8. Le t Q' be the set of ordinals which are less than or equal to the firs t uncountable ordinal Q, w ith the order topology. T h is topology is H 2, S 2 and T 2 but not S 3. Fo r Q' ~ [i2] is sequentially compact without being closed.

E

9. Let (X be (IN w ith a new point c added. Let consist of the open sets of f i N and the complements of finite sets of @N w ith respect to X . The resulting topology satisfies S 0 and H 2 but not S x.

P ro o f. The resulting space is T 4 and a ll sequences taking on an infinite number of distinct values converge to c and to no other point of X ; so ( X , 2Г) is S 0 and H 2. Any side point of a sequence (highly divergent in the space ftN) is a side point of the sequence in X (sequence is con­

vergent in X) . So ( X, #~) does not sa tisfy S 4.

E

10. The free union of Example 9 w ith a copy of (3N is a S 0 without being H 4 or S x.

E

11. The one point compactification of Example 4 satisfies S x and H 2 but not S 2.

I t is interesting to note that there are S 6 spaces that are not T 2.

See Fróchet [6], p. 213.

R eferences

[1] С. Е. A u li, A note on countable paracompact spaces and metrization, Proc.

Am. Math. Soc. 16 (1965), p. 1316-1317.

[2] — A certain class of topological spaces, Prace Matematyczne 11 (1967), p. 49-53.

[3] H. C u llen , Unique sequential limits, Bull. Unione Mat. Ital, III-20 (1965), p. 123-124.

[4] R. M. D u d le y , On sequential convergence, Trans. Am. Math. Soc. 112 (1964), p. 483-507.

[5] S. F r a n k lin , Spaces in which sequence suffice, Fund. Math. 57 (1965), p. 107-115.

[6] M. F r e c h e t, Les espaces abstracts, Paris 1951.

‘[7] S. Graal, Point set topology, New York 1964.

[8] L. G illm a n and M. J e r is o n , Pings of continuous functions, New York 1960.

[9] F. H a u s d o r ff, Gestufte Раите, Fund. Math. 26 (1936), p. 481-502.

[10] J. L. K e lle y , General topology, New York, 1955.

[11] J. K is y ń s k i, Convergence du Type L, Coll. Math. 7 (1960), p. 205-211.

[12] C. K u r a to w s k i, Topologie I , Warsaw (1958).

[13] P. U r y s o h n , Sur les classes (Л?) de M. Frechet, l’Enseignement Math. 25 (1926), p. 77-83.

[14] A. W i l a n s k y , Between T x and T2, American Mathematical Monthly, 74 (1967), p. 266-267.

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