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Topology Proceedings Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: [email protected] ISSN: 0146-4124 COPYRIGHT °c by Topology Proceedings. All rights reserved. TOPOLOGY PROCEEDINGS Volume 28, No. 1, 2004 Pages 1-18 A WEAK ALGEBRAIC STRUCTURE ON TOPOLOGICAL SPACES AND CARDINAL INVARIANTS A. V. ARHANGEL’ SKII Abstract. The notion of !-diagonalizability, introduced in [6], is applied in this paper to the theory of cardinal invariants of topological spaces. A basic lemma stablishes a connection between !-diagonalizability, the ¼-character, and the count- ability of the pseudocharacter in a Hausdorff space. This lemma permits one to prove that every !-diagonalizable lo- cally compact Hausdorff space of countable tightness is first countable, which answers a question asked in [4]. Applications to the study of power-homogeneous compacta of countable tightness are given. In particular, we show that every power- homogeneous locally compact monotonically normal space is first countable. This theorem implies a result of M. Bell in [8]. 1. Introduction Given an algebraic structure (a group structure, a ring structure, a vector space structure, and so on), a standard problem to con- sider is what kind of topologies can be introduced on this structure so that they fit it nicely (make the operations continuous, for ex- ample). It is much more rare that the inverse approach is adopted: given a topological space X, find out if it is possible to introduce 2000 Mathematics Subject Classification. Primary 54A25, 54D50; Secondary 54C35. Key words and phrases. character, ¼-character, point-countable type, power-homogeneous space, sequential space, ¿-diagonalizable space, ¿-twister, tightness. 1 2 A. V. ARHANGEL’ SKII some algebraic structure on this space in such a manner that the structure fits the topology of X well (again, makes the operations continuous, for example). Of course, such algebraic structures must be of a very general nature, if we want them to exist on rather gen- eral (for example, non-homogeneous) spaces. Curiously, it turns out that some such algebraic structures can indeed be defined on many spaces, and, what is more astonishing and important, they can pro- vide effective techniques in situations which, apparently, cannot be treated by purely topological means. The general notion of diag- onalizability, defined under this approach in [5] and [6], has been shown to have applications to extensions of continuous functions, to homogeneity problems, and to completions and compactifications (see [4], [5], [6]). In this paper, we continue to work with the notion of !-diagonalizability, introduced in [6], and provide further appli- cations of it to power homogeneity and to the theory of cardinal invariants. Below, ¿ always stands for an infinite cardinal number. All spaces considered are assumed to be T1. A set A ½ X will be called a G¿ -subset of X, if there exists a family γ of open sets in X such that jγj · ¿ and A = \γ. If x 2 X and fxg is a G¿ -subset of X, we say that x is a G¿ -point in X. In this case, we also say that the pseudocharacter of X at e does not exceed ¿, and write Ã(e; X) · ¿. Recall that the character of a space X at a point x does not exceed ¿ (notation: Â(x; X) · ¿) if there exists a base Bx at x such that jBxj · ¿.A compactum is a compact Hausdorff space. In general, our terminology and notation follow [15]. 2. Cardinal invariants and !-diagonalizability In this section, we give some sufficient conditions for a space to be !-diagonalizable and consider how !-diagonalizability influences relations beween cardinal invariants. We say that the ¼¿-character of a space X at a point e 2 X is not greater than ¿ (and write ¼¿Â(e; X) · ¿) if there exists a family γ of non-empty G¿ -sets in X such that jγj · ¿ and every open neighborhood of e contains at least one element of γ. Such a family γ is called a ¼¿-network at e. If ¿ = !, we use expressions ¼!-character and ¼!-network. In particular, if X has a countable ¼-base at e, then ¼!Â(e; X) · !. A WEAK ALGEBRAIC STRUCTURE 3 The following notion was introduced in [6]. A ¿-twister at a point e of a space X is a binary operation on X, written as a product operation, satisfying the following conditions: a) ex = xe = x, for each x 2 X; b) for every y 2 X and every G¿ -subset V in X containing y, there is a G¿ -subset P of X such that e 2 P and xy 2 V , for each x 2 P (that is, P y ½ V ) (this is the G¿ -continuity at e on the right); and c) if e 2 B, for some B ½ X, then, for every x 2 X, x 2 xB (this is the continuity at e on the left). Proposition 2.1. [6] If Z is a retract of X and e 2 Z, and there is a ¿-twister at e on X, then there is a ¿-twister on Z at e. Proof: Fix a retraction r of X onto Z and a ¿-twister on X at e, and define a binary operation Á on Z by the rule: Á(z; h) = r(zh). Clearly, the operation Á is a ¿-twister on Z at e. ¤ If a space X has a ¿-twister at a point e 2 X, we will say that X is ¿-diagonalizable at e. A space is called ¿-diagonalizable if it is ¿-diagonalizable at every point. It follows from Proposition 2.1 that a retract of a ¿-diagonalizable space is ¿-diagonalizable. We also need the next easy-to-prove statement from [6]: Proposition 2.2. If e is a G¿ -point in a space X, then there exists a ¿-twister on X at e. Proof: Put ey = y for every y 2 X, and put xy = x for every x and y in X such that x 6= e. This operation is obviously a ¿-twister on X. ¤ Theorem 2.3. Let X be a Hausdorff space. Then Ã(e; X) · ¿ if and only if ¼¿Â(e; X) · ¿ and X is ¿-diagonalizable at e. Proof: If Ã(e; X) · ¿, then X is ¿-diagonalizable at e by Proposi- tion 2.2, and γ = feg is a ¼¿-network at e. Therefore, ¼¿Â(e; X) · ¿. Now assume that X is ¿-diagonalizable at e and ¼¿Â(e; X) · ¿. Fix a ¿-twister at e and a ¼¿-network γ at e. Take any V 2 γ and fix yV 2 V . There exists a G¿ -set PV such that e 2 PV and PV yV ½ V . Put Q = \fPV : V 2 γg. Clearly, Q is a G¿ -set and e 2 Q. 4 A. V. ARHANGEL’ SKII Claim: Q = feg. Assume the contrary. Then we can fix x 2 Q such that x 6= e. Since X is Hausdorff, there exist open sets U and W such that x 2 U, e 2 W , and U \ W = ?. Since xe = x 2 U and the multiplication on the left is continuous at e, we can also assume that xW ½ U. Since γ is a ¼¿-network at e, there exists V 2 γ such that V ½ W . Then for the point yV , we have yV 2 W , xyV 2 PV yV ½ V ½ W and xyV 2 xV ½ xW ½ U. Hence, xyV 2 W \ U and W \ U 6= ?, a contradiction. It follows that Q = feg. ¤ Corollary 2.4. Suppose that X is a Hausdorff space !-diagonal- izable at a point e. Suppose further that there exists a countable set A of G±-points in X such that e 2 A. Then e is also a G±-point in X. Corollary 2.5. The space ¯! is not !-diagonalizable at any point e of ¯! n !. Now we can identify many other examples of non-!-diagonal- izable spaces. For example, the Alexandroff compactification of an uncountable discrete space is not !-diagonalizable at the non- isolated point, by Corollary 2.4. Recall that a space X is of point-countable type [1] if every point of X is contained in a compact subspace F ½ X such that F has a countable base of neighborhoods in X. The class of spaces of point-countable type contains all locally compact Hausdorff spaces, all Cech-completeˇ spaces, and all p-spaces. Since the character and the pseudocharacter coincide in Haus- dorff spaces of point-countable type ([1], [15]), the next statement follows from Theorem 2.3. Proposition 2.6. Suppose that X is a Hausdorff space of point- countable type and e 2 X. Then the following conditions are equiv- alent: a) X has a base of cardinality · ¿ at e; and b) X is ¿-diagonalizable at e and has a ¼-base at e of cardinality · ¿. We also need the following statement: Proposition 2.7. If X is a Hausdorff space of point-countable type and the tightness of X is countable, then the ¼!-character of X is also countable. A WEAK ALGEBRAIC STRUCTURE 5 Proof: Take any x 2 X, and fix a compact subspace F of X such that x 2 F and F is a G±-subset of X. Since t(F ) · ! and F is compact, there exists a countable ¼-base ´ of the space F at x (by a theorem of B. E. Sapirovski˘ı[25]).ˇ Every P 2 ´ is a G±-subset in X, since F is a G±-subset of X. Therefore, ´ is a ¼¿-network of X at x. Hence, ¼!Â(x; X) · j´j · !. ¤ Theorem 2.8. If X is an !-diagonalizable Hausdorff space of point-countable type, then the tightness of X is countable if and only if X is first countable.
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