Variations on countable tightness

Topology and its Applications, 2014

Countable tightness may be destroyed by countably closed forcing. We characterize the indestructibility of countable tightness under countably closed forcing by combinatorial statements similar to the ones Tall used to characterize indestructibility of the Lindelöf property under countably closed forcing. We consider the behavior of countable tightness in generic extensions obtained by adding Cohen reals. We show that certain classes of well-studied topological spaces are indestructibly countably tight. Stronger versions of countable tightness, including selective versions of separability, are further explored.

Topology and its Applications, 1993

Arhangel'skii, A.V. and D.N. Stavrova, On a common generalization of k-spaces and spaces with countable tightness, Topology and its Applications 51 (1993) 261-268. Several generalizations of tightness (such that in the countable case it could also serve as a generalization of k-spaces) are defined and studied. Cases in which the new invariants coincide with the usual tightness are obtained.

Transactions of the American Mathematical Society, 1982

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.

Proceedings of the American Mathematical Society, 2017

Extending a result of R. de la Vega, we prove that an infinite homogeneous compactum has cardinality c if it is the union of either countably many dense or finitely many arbitrary countably tight subspaces. The question if every infinite homogeneous and σ-countably tight compactum has cardinality c remains open. We also show that if an arbitrary product is σ-countably tight, then all but finitely many of its factors must be countably tight.

Bulletin of the American Mathematical Society, 1988

One of the most basic and natural generalizations of first countability is countable tightness: the condition that, whenever^ is in the closure of A, there is a countable subset B of A such that x 6 B. Countably tight spaces include sequential spaces, i.e., those in which closure is obtainable by iteration of the operation of taking limits of convergent sequences. The two classes are distinct, since there are easy examples of countable, nondiscrete spaces with only trivial convergent sequences. On the other hand, it was long a major unsolved problem whether every compact Hausdorff space of countable tightness is sequential. First posed in [1] and motivated by the main results of [2], it gained importance from subsequent discoveries on the strong structural properties enjoyed by compact sequential spaces (see [3] and its references). A negative answer was shown to be consistent by Ostaszewski [4], who used Gödel's Axiom of Constructibility (V = L) to construct a countably compact space X whose one-point compactification is countably tight; of course, no sequence from X can converge to the extra point. Now (Theorem 2) we have shown that a positive answer follows from the Proper Forcing Axiom (PFA), introduced in [5]. Our research has uncovered many other striking consequences of PFA, numbered below. None was known to be consistent until now, nor was the following consequence of Corollary 1 and Theorem 3: the ordinal space oj\ embeds in every first countable, countably compact, noncompact Ti space (in particular, in every countably compact nonmetrizable T<i manifold), assuming PFA. This remains true if "first countable" is weakened to "character < ui," meaning every point has a local base of cardinality < u)i. This leads to a remarkable structure theorem for regular, countably compact spaces of character < ui (e.g. the product space [0, l] Wl): under PFA, the closure of every set A can be taken by first adjoining all limits of convergent sequences, and then adjoining to the resulting set A" all points x for which there is a copy W of w\ in A~ such that W U {x} is homeomorphic to ui + 1. Although large cardinals are needed to prove the consistency of PFA, all our PFA results are consistent if ZF is consistent. This is established by using u^-p-i-C-[6, Chapter VIII] posets and a ground model with a o W2-sequence to capture approximations to possible counterexamples in a countable support

Topology and its Applications, 2010

We prove that every LΣ(n)-space (that is, the image of a separable metrizable space under an at most n-valued upper semicontinuous mapping) is a union of n subspaces of countable pseudocharacter and has countable tightness. In particular, every LΣ(n)-space has a dense set of G δ -points, and every LΣ(n)-topological group has countable network.

Journal of Mathematical Analysis and Applications, 2003

It is well known that the space C p ([0, 1]) has countable tightness but it is not Fréchet-Urysohn. Let X be a Cech-complete topological space. We prove that the space C p (X) of continuous real-valued functions on X endowed with the pointwise topology is Fréchet-Urysohn if and only if C p (X) has countable bounded tightness, i.e., for every subset A of C p (X) and every x in the closure of A in C p (X) there exists a countable and bounding subset of A whose closure contains x. We study also the problem when the weak topology of a locally convex space has countable bounded tightness. Additional results in this direction are provided.

Topology and its Applications, 2004

Given a Hausdorff space X, we calculate the tightness and the character of the hyperspace CL ∅ (X) of X, endowed with either the co-compact or the lower Vietoris topology, and give some estimates for the tightness of CL ∅ (X), endowed with the Fell topology.

Glasgow Mathematical Journal, 1999

The purpose of this note is to prove a results of Jain and López-Permouth under a weaker conditions replacing R-weak injectivity by R-tightness and even getting a simpler proof.

Journal of Mathematical Analysis and Applications, 2002

Pfister (1976) and Cascales and Orihuela (1986) proved that precompact sets in (DF)-and (LM)-spaces have countable weight; i.e., are metrizable. Improvements by Valdivia (1982), Cascales and Orihuela (1987) and Kakol and Saxon (preprint) have varying methods of proof. For these and other improvements a refined method of upper semi-continuous compact-valued maps applied to uniform spaces will suffice. At the same time, this method allows us to dramatically improve Kaplansky's theorem, that the weak topology of metrizable spaces has countable tightness, extending it to include all (LM)-spaces and all quasibarrelled (DF)-spaces, both in the weak and original topologies. One key is showing that for a large class G including all (DF)-and (LM)spaces, countable tightness of the weak topology of E in G is equivalent to realcompactness of the weak * topology of the dual of E.