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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 48, Number 2, April 1975 SPACES IN WHICHCOMPACT SETS HAVE COUNTABLE LOCAL BASES RALPH R. SABELLA ABSTRACT. spaces D -spaces in which Among cient compact the results condition a question related under posed sufficient they be coconvergent. spaces; for such paper neighborhood spaces to such set that of natural numbers and Jlix) fot all Ix 72. Using the notation j, we define CGplyJ there a space iCplxJ'D is an ONA "U is P." CplyJ) U having is that in stratifiable is metrizable lame if it there tnere is a function X is a topological is the open space, neighborhood Sy^i is U-linked system of x. points {contraconvergent) whenever \yn ! is (Minkedto property N is the to |xnS if y S for the set of cluster coconvergent some to prop- |Jl(x): x £ X\ Cpíx to be a topological condition. (ONA) x £ lj{x, n) = L¡n{x), where If U is an ONA then the sequence is a suffi- developable a D .-space of In relation metrizability a nesting-like U: Xx N-*\J paper that be implies assignment examples (Dg-spaces). coconvergent. shown that satisfying are bases in this are it is Coconvergence in this local given spaces semimetrizable it is shown spaces countable F. B. Jones, a stratification An open coconvergent have to D.-spaces which by erty exists and sets P, we shall of if Cp\x |xj. say £ U ix ) "X \ If on X is P" or If xQ is a limit point of S*„!, then M*„> xQ) ~ ^xk: & = 0, 1, 2, ..-}. Coconvergence compact sets. Proposition can be characterized From Received of countably based [6] we have 1. X is coconvergent that for any compact which Uit^U): in terms set iff there K and open is an R containing ONA U on X such K there is a k £ N for x e K^ c Rby the editors AMS (MOS) subject October classifications Key words and phrases. coconvergent, contraconvergent, 10, 1973 (1970). and, in revised form, January 31, 1974. Primary 54E35; Secondary Open neighborhood assignments, stratifiable spaces, Nagata (/-linked spaces. 54D99- sequences, Copyright © 1975. American Mathematical License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 499 Society 500 r. r. SABELLA A D -space is defined F has by C. E. Aull closed set U (F) is an open set containing a countable U .{F) C R for some Definition 1. k £ N. local base is in which \U (F)S, i.e. for each n F, and if F C R, where Remaining A space [l] as a space with DQ if each Aull's every n £ N, R is open, nomenclature compact then we introduce set has a countable local base. Without loss for a compact T-., D.-spaces, countable of generality set as well [l] and, A first hence, countable spaces, countable are spaces is equivalent Ix^S is U-linked to the constant Proposition 2. X is a D^-space iff for each Proof. K. of open \x n \ clusters Say sets with |i7fe(K)S a local n, x^ £ UniK) but Uk\ does is an 772£ N such that for all n > m, x and for some D0-spaces. is an ONA sequence that Sx! then characterization: compact K such Point- there of this containing base countable; K C X there if x £ U (k) is a for in K. X is a DQ-space If for all D0-spaces. are also is an analogue local is first to one on which The following \U^{K)\ a given DQ-space to x. n then set second space assume every converges sequence all Trivially, as coconvergent U such that whenever ixn\ we will always to be nested. base not cluster 4 K. Hence, for the compact in K, then KCX-clix k > m, Uk{K) C X - cl Sx^: 72> ttzS, which there ; 72> ttz!, contradicts the condi- tion that xk e UAk). Conversely, contained x £ K, x assume in the open 6 Cpjx^S. the condition set to be satisfied. R, and if x But R £jl{x) If K is compact £ (J (K) - R for all and contains none n, then of the and for some x , which is a contradiction. In coconvergent tion spaces 1 is equivalent the characterizing to the existence condition of an ONA V such given that in Proposifor each AUn,*0), iUfV„U): x £ Mxk, x0)SS~=1 is a local base for A(x , x ) [6]. As the next proposition exists Proposition each and following example show, no such equivalence for DQ-spaces. 3. A(x , xQ) has Proof. choosing Let X be a T ^space. a countable The sufficiency a nested first part countable local Then X is first countable The converse follows iff base. is immediate. ONA U on X and noting License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use that, by for any SPACES IN WHICH COMPACT SETS HAVE COUNTABLE BASES M*n, x0) follows and each that Iß, k, there .(A(x is an , x0))i?" nk such that . j is a local xn £ UkixQ) base for Mx^, fot 501 n> n It x ) where k S*./(A(V U £//*,) X0]) = Uk{x0) U r =l The following which is an example of a first countable, stratifiable space is not a D.-space: Let X = f(x, y) £ R2: y > OÍ. Define U by (Ma, b) = |(a, y) £ X: \y- b\< l/n\ ¡7n(a, 0) = |(x, y) £ X: y < l/n, U defines a first countable, x < lS is compact. Tj Assume ii b > 0. 0 < |x - a\ < l/n\ topology on X. it has a local base U \{a, 0)\. The set K = S(x, o): 0 < JBR(K)S and let L = !(x, y): y > OS. For each 72 let Kn = |(x, 0) £ K: L%n ß„(K) = 0 S. It follows that K = U integer o 777and an interval is a ze such that .iZa l¡kic, . By Baire's category theorem S(x, O): c _< x _< d\ in which o) C BmiK) and there are an Km is dense. a x 4 c for which There (x , 0) £ tVjc, 0) nKm. Hence, (*, l/2/fe) £ Í7¿(c, O) C Bm(K), i.e. L7 nBmiK)4 0, contradicting (x, 0) £ Km. A space DQ-spaces than is we require sets) The above provides have example being compact normal the weaker countable us with an answer the countably X is not a DQ-space. D^ if it is perfectly where closed Hence local stratifiable, bases, that compact compact is perfect and hence in the negative. condition and countably condition normality perfectly However, [l]. sets normal sufficient? [2], we are able to rim compactness, i.e. spaces each x £ X and U e Tiix) there is a V £ ?I(x) such that For (rather to weaken in which for V C U and bdry V is compact. Proposition compact set Proof. open 4. is compact and rim compact, it is D iff every is a G... Let K be compact and l¡nÍK) be a nested and contains and open // X is regular sets K, there V(xj), ... and contained is a finite , V(x^) with in R. Hence, number G g of K. If R is of elements V(x¿) £ ?í(x¿) such B = bdry(U*Lj License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use of K, x,, ... , x that V{x .) V(x.)) bdry is compact. 502 R. R. SABELLA The sequence |X - cl Un{K)\ covers X - U¿_iV(x-) ar"i, hence, B. It follows that there is an 72£ N such that B C X - cl U ifd and, therefore, Un{K) CR. The converse Lemma 1. is true for any The following (a) X is coconvergent. (b) X is a D^space, D.-space and follows from Proposition 2. are equivalent: each compact set K having a local base III {K)\ such that if X, C K2, both compact, then Uk{Ky) C UkiK2) for all k. (c) X is \Uki&{xn, a space 1 xQ))\ such in which that each A(x 22 , xAU Ukix) C L/fc(A(xn, x0)) has a local base for each x £ A(xn, xQ) awfi ONA on X. For each compact a// zL Proof. Let U be a nested K let UniK) = \J\Unix): x £ XS. By Proposition for X. It follows immediately C Un{K2) tot all Since A(xn, xQ) is compact, paragraph preceding noted in there exists lUiV (x) : x £ A(xn, an ONA to the existence that (c) implies (a). Lemma Let coconvergent Proof. Since Without loss sequence § that X such that is a local follows base 3, for then ^„(X,) Then from (b). the any As condition A(x that , x ), for A(x , x0) is equivalent ONA on X. X is semistratifiable, It easily follows X is developable of generality is a development for all contradicts R. W. Heath D0-space from this if it is an ONA U which \j in [5] gives 72 let X, for if there are sequences ix that if Ix^S converges For each for 72 there is an ONA V such to Sx^j, then we may choose = \Unix); is semi- x £ X!. is an x £ X and \ and to x |yn! The R £ for which |x! £ U (x ) - R, it would follow that x £ Cplyn!, the way in which which there Ix! is V-linked is (i-linked to |x S and y able), (c) trivially Proposition X be a T^-space. and coconvergent. Jl(x) which on XqMS^j sequence stratifiable such is a local base and semistratifiable. the constant [4]. V of a coconvergent 2. 1, \UniK)\ that if Xj C X2, both compact, 72. each the coconvergent \yA an example is not developable. to a conjecture by F. B. Jones, semimetrizable space was chosen. of a semimetric The example viz. a topological to be a developable (hence space was in answer property is that License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use semistratifi- sufficient it be DQ. for a Lemma 2 SPACES IN WHICH COMPACT SETS HAVE COUNTABLE BASES shows that coconvergence is not known whether, is one in fact, topological property coconvergence which is not also will 503 work. a necessary It is condi- tion. Proposition 5. X is a Moore space if it is regular, rim compact and semistratifiable. Proof. Let U be a semistratifiable [>J\U (x); x £ XÎ for any compact \U, (X)S is a Gg , and using X. Now by Lemma Proposition ification 6. The last spaces. But then We have is sufficient partial set also are applications 7. X is metrizable 8. Proof. for X in X. of D „-spaces to stratifiable space and coconvergent. Proposition of Proposition 3: not every 7 be weakened, be DQ to be metrizable? X is metrizable X having a local if it is stratifiable base stratifiable viz. We have 7 give and D. is it only a with each \X - (X - X)~ S. (V/e are using stratification of the open If Xj and X2 are compact with Proposition implies in X* and has a local base iff it is stratifiable following (X - X.)I22 for all 72 and, hence, Proposition n XS is a local of X, is a [7] we have an increasing Proof. compact- in Proposition denote and its one point in X, it is compact that a stratifiable compact for X is a DQ-space. Can the conditions answer T base 2, it is developable. compactification from the example D 2 that 4, it is a local 4, X* the one point !t/„(X) U (X) = from Proposition and by Lemma compact, then two propositions From Proposition space of Proposition normal, If X is compact \Un{K)\. the proof // X is locally By Proposition D0-space. base X, it follows 7, X is coconvergent, is perfectly Proof. ONA on X. Letting set set \R ! to R.) X, C X2, then (X - KA C U (X.) C Un {KA, which byJ Lemma 1 and ¿ * '221 metrizability. 9. // X is stratifiable By Proposition paracompactness and rim compact, 5, X is a IVloore space. which is sufficient it is metrizable. But stratifiability in Moore spaces ity. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use for metrizabil- 504 R. R. SABELLA REFERENCES 1. C. E. Aull, Closed set countability axioms, Ser. A 69 = Indag. Math. 28 (1966), 311-3162. C. J. R. Borges, On stratifiable Nederl. Akad. Wetensch. Proc. MR 33 #7973- spaces, Pacifie J. Math. 17 (1966), 1—16- MR 32 #64093- J. G. Ceder, (1961), 105-1254. G. D. Creede, (1970), 47-54. 5. 6. of metric spaces, Pacific J. Math. 11 Concerning semi-stratifiable spaces, Pacific J. Math. 32 MR 40 #8006. R. W. Heath, 60, Princeton Some generalizations MR 24 #A1707. On certain Univ. Press, R. R. Sabella, first-countable Princeton, Convergence N.J., spaces, Ann. of Math. Studies, no. 1965, pp. 103—113- properties of neighboring sequences, Proc. Amer. Math. Soc. 38 (1973), 405-409. 7.-, Properties of neighboring sequences in stratifiable spaces, Proc. Amer. Math. Soc. (submitted). DEPARTMENT OF MATHEMATICS, CALIFORNIA STATE UNIVERSITY, NORTHRIDGE, CALIFORNIA 91324 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use