Products of ¿-spaces and Spaces of Countable Tightness

In this paper, we obtain results of the following type: if /: X -» Y is a closed map and X is some "nice" space, and Y2 is a &-space or has countable tightness, then the boundary of the inverse image of each point of Y is "small" in some sense, e.g., Lindelöf or «¿¡-compact. We then apply these results to more special cases. Most of these applications combine the "smallness" of the boundaries of the point-inverses obtained from the earlier results with "nice" properties of the domain to yield "nice" properties on the range. Introduction. Recall the following theorem due to Morita and Hanai [14] and Stone [17]. Theorem. // /: A -» Y is closed and X is metrizable, then the following are equivalent. (a) Y is first countable; (b) For each y E Y,df~x(y) is compact; (c) Y is metrizable. The (c) =» (b) part is due to Vaïnsteïn [22]. But even the (a) => (b) part holds under much more general conditions: Michael [7] showed (b) holds if A is paracompact, and Y is locally compact or first-countable. Note that the assumptions on Y in Michael's theorem could not be weakened to "Fis a ¿-space" or "F has countable tightness": the map identifying the limit points of a topological sum of k convergent sequences is a closed map from a metrizable space A to a Fréchet space Y, and | d/"'( y) | = k for some y E Y. In this paper, we show that the situation is different if we require Y2 to be a ¿-space or have countable tightness. (Recall that the square of a ¿-space or a space of countable tightness need not have the same property.) We will usually not be able to show that the boundaries of point-inverses are compact, but we will often (depending upon conditions imposed on A or Y) he able to show that they are "small" in some sense, e.g., Lindelöf or u,-compact. In the second section, we apply general results of this type to more special cases, often combining the "smallness" of the boundaries of point-inverses with "nice" properties of A to obtain "nice" properties of Y. We mention the following earlier result of the second author [21] which is related to this topic. Received by the editors February 11, 1980 and, in revised form, July 13, 1981. 1980 Mathematics Subject Classification. Primary 54D50; Secondary 54A25, 54C10, 54D55. ©1982 American Mathematical Society 0002-9947/81 /0000-0614/$03.50 299 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 300 G. GRUENHAGE AND Y. TANAKA Theorem. // /: A -» Y is closed and X is metrizable, then the following are equivalent. (a) For each y E Y, df'x(y) is Lindelöf. (b) Y has a point-countable k-network [see §2, Definition 2.1]. (c) Y has a a-locally-countable k-network. See [7] and [21] for other related results. We will often make use of the following well-known property of closed maps (cf. [3, p. 52]): If /: X ^ Y is closed, then for each y E Y and open U E X such that /"'(y) E U, there is a neighborhood V of y such thatf~x(V) E U. 1. General results. All our spaces are assumed to be regular and Tx. We consider cardinals to be initial ordinals. We now recall some basic definitions. Definition 1.1. A space A has the weak topology with respect to a collection G of sets if a subset A of A is closed (resp., open) in A if and only if A D C is closed (resp., open) in C for each C E G. Definition 1.2. A space Ais a k-space (quasi-k-space) if A has the weak topology with respect to its compact (countably compact) subsets. A is sequential if A has the weak topology with respect to its compact metric subspaces (equivalently, with respect to its subspaces homeomorphic to co + 1, a sequence with its limit point). A has countable tightness (denoted by f(A)^w) if it has the weak topology with respect to its countable subsets. We will be using the following elementary facts about these concepts. (i) If A has the weak topology with respect to a collection G, and /: A -> Y is a quotient map, then Y has the weak topology with respect to {/(C): C E Q). Thus all properties named in Definition 1.2 are preserved by quotient maps. (ii) If A satisfies any of the properties in Definition 2.2 locally, then the whole space has the property. (iii) If A has a locally finite cover by a family G of closed sets, then A has the weak topology with respect to Q. Definition 1.3. A space A is (strongly) collectionwise Hausdorff if whenever {xa: a E A} is a closed discrete subset of A, there exists a (discrete) disjoint collection {Ua: a E A) of open sets such that xa E Ua for each a E A. Note that every normal collectionwise Hausdorff space is strongly collectionwise Hausdorff. Let c denote the cardinality of the continuum. Theorem 1.4. Suppose f: X -> Y is closed, with X strongly collectionwise Hausdorff. Then the boundary, df'x(y), off~x(y) is c-compact for each y E Y if either (a) Y2 is quasi-k and t(Y) *z u or (h) t(Y2)< w. Proof. Suppose 3/"'( v) is not c-compact. Then there is a closed discrete subset D E df~x(y), with | D | = c. For each d ED, let U'd be an open set containing d such that {U'd: d E D) is discrete. Let d E Ud E Ud E U'd, where Ud is open. Note that {Ud: d E D) is also discrete. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use PRODUCTS OF ¿-SPACES 301 For each d E D, y Ef(Ud — f x(y)). Since t(Y) « co, there is a countable set {ydn: n E co} Ef(Ud~f-\y)) such that y E {ydn: n_E co}. Let A,„ = /-'(^,„) n t/„, and let A„ = /"'(y) n £/„. If O is open and contains Xd, then there is an open set O' such that O' n t/¿ = 0, and /_,(>>) COUO'. Let W be the complement in Y of /( A (O U O')). Then V E W, so there is n E « such that yd „ E W, and hence f~x(ydn) COUO'. Thus A¿ „ C(OU 0')nÜd= OH ÜdCO. Choose xd „ E A^„, and letyl^ = {xd„; n E co}. By the above argument, every open set containing Xd contains infinitely many elements of A d. For x E Ad, let Dx = {d' E D: there exists x' E Ud, with f(x) =f(x')}. Let Bd — {x E Ad: Dxis uncountable}. Claim 1. Xd n Bd — 0. To see this, let Bd = [x0, xx,...}. Inductively choose a sequence d0, dx,... of distinct elements of D, and points x'n E Ud such that f(x'„) =f(x„). Then {x'0, x\,...} is a closed subset of A, so f({x'0, x\,...}) = f(Bd) is closed. Thusy £ /(/>J =f(Bd) = /(//¿),_so A¿ n Bd = 0. Let Cd = A_dBd. By Claim 1, XdnCd¥= 0. Pick ¿(0) E D. Let 7)(c/(0)) = U {Dx; x E Cd(0j}. Observe that D(d(0)) is countable. If d(ß) has been chosen for all ß < a < c, let d(a) EDUß n Q(a) ^ 0, so v E/(Q(a)) = /(Cd(a)). For each n E co, choose x„ E Q(n) such that xj E O. Then {x„: /!£«}=£„ for some a, and ea„ = xn for each «. There is n E co such that c\ „ E O. Thus (e*„, c* n) E O2 fl 7/a, which proves the claim. The next claim completes the proof of part (a). Claim 3. If K E Y2 is countably compact, then K n H is finite. To see this, suppose a0, a,... are distinct ordinals such that for each n E to, K n Ha ¥= 0. Then we can find (e*nkn, c*a k ) E K n // . But (cx ^ : n E co} is a closed discrete subset of A, since cx k E {/¿^ ,. Thus {(e* ft , c*o k ); n E co} is an infinite closed discrete subset of K, contradiction. Thus K meets only finitely many //a's. Now suppose that for fixed a, K n Ha is infinite. Then for each n E co, we can find (e*k ,c*k)EKn Ha. But [eak : n E co} is an infinite closed discrete subset of A and we get a contradiction as before. Thus each K n Ha is finite, and so K n H is finite. To complete the proof of part (b), we have the next claim. Claim 4. No countable subset of H contains (y, y) in its closure. Suppose C E H, |C| n. Also recall cx^ E Cd(X ) E Ud(Xa y Thus cXa n E Vm, which means n > m, a contradiction. This proves Claim 4. Hence Y2 does not have countable tightness, a contradiction which proves the theorem. D Assuming the continuum hypothesis (CH), we have the following corollary. Corollary 1.5 (CH). Suppose f: X -» Y is closed, with X paracompact. Then each df~x(y) is Lindelöf if either Y2 is a k-space with t(Y) < co, or t(Y2) «£ co. Proof. Immediate from Theorem 1.4 and the fact that co,-compact paracompact spaces are Lindelöf [1]. Remark. By the proof below, if Y2 is a ¿-space with t(Y) « co, then t(Y2) < co. Thus the two conditions are not independent. Proof. Since Y2 is a ¿-space, it has the weak topology with respect to the collection of compact subsets of Y2; that is A E Y2 is closed whenever A E C is closed in C for every compact subset of C of Y2. Each compact subset C of Y2 is contained in ir(C)2, where it is the projection from Y2 onto Y. Then Y2 has the weak topology with respect to {tr(C)2; C is compact in Y2}. Since each w(C) is a compact space of countable tightness, by a result of V. I. Malyhin [5, Theorem 4], so is each 77(C)2. Then i(y2)< co. D We do not know if Corollary 1.5 is true without CH. The problem seems to hinge on strengthening the conclusion of Theorems 1.3 and 1.4 by replacing "c-compact" with "co,-compact". It turns out if we add the condition "Y is sequential" to the hypotheses of these theorems, then we can do it. Theorem 1.6. Suppose f: X -» Y is closed with X strongly collectionwise Hausdorff and Y sequential. Then each df~x(y) is ux-compact if either Y2 is a quasi-k-space or t(Y ) =£ co. Proof. Suppose Y2 is a quasi-¿-space. Since Fis sequential, by [18,Theorem 2.2] Y2 is sequential, hence t(Y2) < co. Thus we can assume that t(Y2) < co. Suppose 3/ '(j) is n°t Y closed with A locally compact and paracompact, such that Y2 is a ¿-space, but 3/~'( v) is not Lindelöf for some^ E Y. Proof. For each a < co,, let S(a) he a copy of ordinal space co, + 1. Let A be the free union of (S(a): a < co,}. Let T be the space obtained from A by identifying the point co, in each copy to a single point oo. Let/: A -> Y he the quotient map. Then Ais paracompact and locally compact,/is closed, and 3/"'(oo) is not Lindelöf. Ais a ¿-space (being locally compact), hence so is Y. It remains to prove that Y2 is a ¿-space. First we introduce some notation. For each a, ß «s co,, let ß(a) be the image under/of the element of S(a) corresponding to the ordinal number ß. If ß < ß' < co,, let [ß(a), ß'(a)] = {y(a): ß < y « /?'}, and let [ß(a), oo] = [¿8(a), co,(a)]. Suppose A E Y2, with A ¿-closed, but not closed. Since for each a, ¿8 < co,, [0(a), ¿8(a)] X Y and YX [0(a), ¿3(a)] are clopen ¿-subspaces of Y2, it must be true that (oo, oo) E A — A. Since [f(S(0))]2 n A is closed, there exists y0 < co, such that [y0(O), oo]2 n A = 0. Now suppose ya has been defined for all a < ¿8, where ¿8 < co,, in such a way that the following property Pa holds. Pa(0i(a,),&(«2)) EA and «i. <*2 ̂ « implies /?, < ya, or ß2 yß. Then it must be true that ¿8, < 8a ß (by (ii) above). If ya < ßx < 8aß, then since yß > ôfo, we have (¿8,(a), ß2(ß)) E ({ßx(a)} X [8^ß(ß), oo]) n A, a contradiction. Thus ¿8, < ya, so Pß holds. Thus we can define {ya: a < co,} in such a way that Pa holds for each a < co,. Let U= {¿8(a): ß > ya, a < co,}. Then U is an open set in Y containing oo. Since (oo, oo) E A, there exists (¿8,(a,)¿82(a2)) E U2 D A. Since Pa +a holds, either ßx < ya or ¿82 < ya . But then either ßx(ax) E U or ß2(a2) E Í7, contradiction. Thus F2 is a ¿-space. 2. Applications. As applications of results in §1, we shall consider the products of ¿-spaces and spaces of countable tightness in more special cases. Definition 2.1 [8,16]. A collection 9 of (not necessarily open) subsets of a space A is a k-network for A if, whenever CEU with C compact and U open, then C C U f C U for some finite subcollection § of 9. An espace is a space with a a-locally finite ¿-network, and an X0-space is a space with a countable ¿-network. Note that metrizable spaces are X-spaces, and separable metrizable spaces are N0-spaces. We say that A is a locally tf0-space if each point of A has a neighborhood which is an S „-space. Theorem 2.2 (CH) Let f: X -» Y be a closed map. Let X be a paracompact, locally KQ-space. Then the following are equivalent. (a)t(Y2) ^ co; (b) each df'x(y) is Lindelöf; (c) Y is a locally K0-space; and (d) Y is locally separable. Furthermore, if Y is sequential, then the CH assumption can be omitted. Proof, (a) => (b): This is Corollary 1.5. (b) => (c): Since each subset of a locally K0-space is locally S0, as in the proof of [7, Corollary 1.2], we can assume that each/"'(>>) is Lindelöf. Thus,/is a closed map with eachf~x(y) Lindelöf. Then, for each y E Y, there is a closed neighborhood W of y in Y, and open subsets V: of A which are S0-spaces such that f~x(W) E U^LXV,. Since U°l,Pj is an N0-space, so is f~x(W). Since the closed image of an S0-space is also S0 by [8, G], M^ is an N0-space. This implies (c). (c) =» (a) and (c) =» (d): By [8, F], F2 is a locally S0-space. Then, by [8, D, E] F2 is locally a hereditarily separable space. Hence t(Y2) < co. (d) =» (b): This follows from [21, Proposition 1], because F is paracompact, hence is locally Lindelöf by (d). From Theorem 2.2 and some results in [21], we have Corollary 2.3. Let f: A — Y be a closed map with X locally separable metric. Then the following are equivalent. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use PRODUCTS OF ¿-SPACES 305 (a)r(F2) Y be a closed map with each 3/~x(y) Lindelöf. If X is bi-k and Y satisfies /i(S0), then Y is a locally ku-space. Proof. Since each closed subset of A is a bi-¿, as in the proof of [7, Corollary 1.2], we can assume that each f~\y) is Lindelöf. Let .y E F. Then we will prove that each point of f~x(y) has a neighborhood contained in the inverse image of some compact subset of F. To see this, suppose not. Then there is a point a0 of f~x(y) such that for every neighborhood F of a0 and for every compact subset K of F, V GL fx(K). Let %= {A f~x(K); Kis compact in Y). Then °Jis a filter base accumulating at the point a0. Since A is bi-¿, there exists a ¿-sequence (A„) in A such that a0 EFnAJor all n E co and all F E §. Obviously, (f(A„)) is a ¿-sequence in Y. Thus, by condition K(K0), some f(A„n) is compact. Let Ka= f(An/). Then, a0 E(X f~x(K0)) n A„a E (A r'Uo)) n f-x(K0) = 0. This is "a contradiction. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 306 G. GRUENHAGE AND Y. TANAKA Thus, each point x of fx(y) has a neighborhood Vx which is contained in the inverse image of some compact subset of F. Since/"'(.y) is Lindelöf, {Vx: x E f~x(y)} contains a countable subcover {F„}„e„ of f~x(y). For each n, let Kn be a compact subset of F such that Vn E f~x(Kn). Since/is closed and F is regular there exists a neighborhood W of Y such that/"'(W) E Un(EuV„. Let F = f~x(W) and T= {fn Fj-: / E co}. Then, since Tis an open covering of F, F has the weak topology with respect to °V. Since Fn V, Q 'F ñ f~l(K¡) for each / E co, F has the weak topology with respect to {F D f~\Ktf, i E co}. Since/| Fis closed, hence quotient,/(F) = W has the weak topology with respect to {W nK,: i E co}. Thus IF is a ¿„-space, and so F is a locally ¿„-space. D Lemma 2.9. Let f: X -» Y be a closed map with X normal and t(Y) < co. // Y2 is a k-space, then either Y satisfies condition AT(K0) or each 3/~'(y) is countably compact. Proof. According to [20, Theorem 4.2], if the product of two spaces is quasi-¿, and one factor is not an inner-one A -space in the sense of E. Michael, R. C. Olson and F. Siwiec [11], then the other factor satisfies K(a), where a is its tightness. F satisfies condition /£(N0), or F is an inner-one /I-space. If F is inner-one A, by [10, Theorem 9.9] each 3f ~x(y) is countably compact. Lemma 2.10 [10]. Bi-k-spaces are preserved by perfect images and countable products. By invoking Corollary 1.5, and Lemmas 2.7, 2.8, 2.9 and 2.10, we obtain the following theorem. Theorem 2.11 (CH). Let f: X — Y be a closed map with X paracompact bi-k. If t(Y) < co, then the following are equivalent. When Y is sequential, the CH assumption can be omitted. (a) Y2 is a k-space. (b) Y is locally ¿„, or each df'x(y) is compact. (c) F is locally ¿„, or bi-k. Corollary 2.12. Let f: X -» Y be a closed map with X or Y sequential. Let X be a paracompact space of pointwise countable type. Then Y2 is a sequential space (equivalently, a k-space by [18, Theorem 2.2]) if and only if Y is locally ¿„ or bi-k. Before proceeding with the next lemma, we remind the reader that the perfect image of an K-space is an N-space, but the closed image of a locally compact metric space need not be S-space (cf. [21, Theorem 7]). Lemma 2.13. Let f: X -> Y be a closed map with each dfx(y) Lindelöf. If X is an espace, and Y satisfies condition K(K0), then Y is also an espace. Proof. Let 9= U°l,iP, be a a-locally finite ¿-network for A satisfying the following conditions: Each element of P is closed, 9, E ÍP + , and 9, is closed with respect to finite intersections. Let K be an arbitrary compact subset of Y. Since each subset of an S-space is an K-space, as in the proof of [7, Corollary 1.2], we can assume that each /"'( y) is Lindelöf and that there exists a compact subset C of A with/(C) = K. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use PRODUCTS OF ¿-SPACES 307 Let 9' = {P E 9: 9 D C ¥= 0}, and let G he the collection of finite unions of elements of 9' which contain the compact subset C. Then G is a nonempty, countable collection in A. Let G = {P¡: i E co} and C„ — T= i-P,-Ior eacn "• Then (C„) is a ¿-sequence for C. Since (f(Cn)) is a ¿-sequence for K, by 7i(K0) there exists a compact subset f(C„ ) of F. On the other hand, by the conditions of the collection 9, each Cn can be expressed as a union of finitely many elements of 9. So, the compact subset f(C„) containing K can be expressed as a union of finitely many elements of /( 9 ). Let % = {/(P)'P £ ^ and f(P) is compact in F}, and let %* he the union of all elements of %,. Then, since f(9,) Ef(9,+ X), by the above, each compact subset of F is contained in some %*. We will now prove that F is an K-space. Each %, is a hereditarily closure-preserving collection of compact subsets of Y, that is, whenever a subset K' of K is chosen for each K E 9C(-, the collection {/C: it E 5C,} is closure-preserving. This is because %, is the image of a locally finite, hence hereditarily closure-preserving, collection under a closed map. Then by a result of Michael [6, Theorem 1], each %* is paracompact. Next, to see each %* is locally K0, let 91,= {P E 9,: f(P) E%,} and let 91* = U 91,. Then 91* has the weak topology with respect to the locally finite closed collection %,. Also, f\ 91* is closed, hence quotient. Thus %* =/(9l*) has the weak topology with respect to %,. Since/is closed and each f~x(y) is Lindelöf, %, is locally countable. Hence each %* is a locally ¿„-space. Since each compact subset of Ais an K0-space, by [8, G] each compact subset of Fis also K0 because it is the image of a compact subset of X. Then each %* is a locally K0-space, since each point has a neighborhood which has the weak topology with respect to a countable collection of compact K „-spaces (see [8]). So, each %* is a paracompact, locally K „-space. It follows that each %* is also an K-space. As is seen, each compact subset of F is contained in some %*. Since each 9C* is an K-space, it follows that F is also an K-space. This completes the proof of the lemma. Lemma 2.14 [19, Theorem 3.1]. Let Y be a kand X-space. Then Y2 is a kand espace if and only if Y is metrizable, or Y has the weak topology with respect to a countable covering of closed and locally compact subsets of Y. Let a ¿-space F be the closed image of an K-space. Since each closed subset of an K-space is easily seen to be a G8-set, each point of F is a Gs-set. Thus by [10,Theorem 7.3], Fis sequential. Therefore, by Corollary 1.7, and Lemmas 2.13 and 2.14, we have Theorem 2.15. Let f: X -» Y be a closed map with X a paracompact espace. Then Y2 is a k-space if and only if Y is metrizable, or Y is an espace having the weak topology with respect to a countable covering of closed and locally compact subsets of Y. Remark. Let A be an K-space each of whose countable (resp. uncountable) subset has an accumulation point. Then A is an K0-space, and so A is compact (resp. Lindelöf). Thus, by Theorem 1.6, we have the following. If an K-space A is more generally strongly collectionwise Hausdorff, then the statement of Theorem 2.15 is also valid. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 308 G. GRUENHAGE AND Y. TANAKA