Angelo Bella

Spaces which are generated by discrete sets

Abstract Continuing the study initiated by Dow, Tkachenko, Tkachuk and Wilson, we prove that countably compact countably tight spaces, normed linear spaces in the weak topology and function spaces over σ-compact spaces are discretely generated. We also construct, using [CH], a compact pseudoradial space and a pseudocompact space of countable tightness which are not discretely generated.

Tight points and countable fan-tightness

Abstract The countable spaces whose product with the sequential fan S c have countable tightness are characterized. As a consequence, it is shown that if X × S c has countable tightness then X has countable fan-tightness.

Sequential + separable vs sequentially separable and another variation on selective separability

A space X is sequentially separable if there is a countable D ⊂ X such that every point of X is the limit of a sequence of points from D. Neither “sequential + separable” nor “sequentially separable” implies the other. Some examples of this are presented and some conditions under which one of the two implies the other are discussed. A selective version of sequential separability is also considered.

Pseudoradial spaces: Separable subsets, products and maps onto Tychonoff cubes

Abstract We work around our question of whether compact non-pseudoradial spaces have separable such subspaces. We obtain results about products of pseudoradial spaces and obtain more conditions which guarantee that each compact sequentially compact space is pseudoradial. We also discuss some questions of Sapirovskii which are also directed at separating the non-separable from the separable. We reinforce the need to focus on the space I ω 2 .

On R-monolithic spaces

Abstract The class of R-monolithic spaces is properly contained in the class of pseudoradial spaces and includes all sequential spaces, all LOTS and all compact monolithic spaces. We study various properties of these spaces. For instance, it is shown that Changs Conjecture reduces the existence of a compact radial non-R-monolithic space of density not exceeding ℵ 1 to the existence of a separable space of the same type. Furthermore, we prove that the existence of a compact ccc R-monolithic non-sequential space is undecidable in ZFC and that the class of compact R-monolithic spaces is countably productive.

Few remarks and questions on pseudoradial and related spaces

Abstract Some new results concerning pseudoradial and related spaces are presented. Particular emphasis is given to the class of semiradial spaces. In addition, some known facts are collected in order to provide motivations for various questions.

Sequential separability vs selective sequential separability

A space X is sequentially separable if there is a countable D  X such that every point of X is the limit of a sequence of points from D. We present two examples of a sequentially separable space which is not selectively sequentially separable. One of them is in addition countable and sequential.

A further strengthening of a theorem of Juhász and Spadaro

Abstract By using a stronger notion of free sequence, we improve a result of Juhász and Spadaro on the cardinality of a chain of spaces.

Around tight points

Abstract We present three examples of countable spaces with a single nonisolated point. The first gives, assuming CH, a Frechet-Urysohn tight point which does not have countable absolute tightness; the second, constructed with the help of ◊, gives a tight non-weakly Frechet-Urysohn point without countable absolute tightness; the third gives a weakly Frechet-Urysohn point which is not Frechet-Urysohn, has countable fan-tightness but is not tight.

On cardinality bounds involving the weak Lindelöf degree

Abstract We give a general closing-off argument in Theorem 2.3 from which several corollaries follow, including (1) if X is a locally compact Hausdorff space then |X| ≤ 2wL(X)ψ(X), and (2) if X is a locally compact power homogeneous Hausdorff space then |X| ≤ 2wL(X)t(X). The first extends the well-known cardinality bound 2ψ(X) for a compactum X in a new direction. As |X| ≤ 2wL(X)χ(X) for a normal spaceX[4], this enlarges the class of known Tychonoff spaces for which this bound holds. In 2.12 we give a short, direct proof of (1) that does not use 2.3. Yet 2.3 is broad enough to establish results much more general than (1), such as if X is a regular space with a π-base ℬ such that |B| ≤ 2wL(X)χ(X) for all B ∈ ℬ, then |X| ≤ 2wL(X)χ(X). Separately, it is shown that if X is a regular space with a π-base whose elements have compact closure, then |X| ≤ 2wL(X)ψ(X)t(X). This partially answers a question from [4] and gives a third, separate proof of (1). We also show that if X is a weakly Lindelöf, normal, sequential space with χ(X) ≤ 2ℵ0, then |X| ≤ 2ℵ0. Result (2) above is a new generalization of the cardinality bound 2t(X) for a power homogeneous compactum X (Arhangelskii, van Mill, and Ridderbos [3], De la Vega in the homogeneous case [10]). To this end we show that if U ⊆ clD ⊆ X, where X is power homogeneous and U is open, then |U| ≤ |D|πχ(X). This is a strengthening of a result of Ridderbos [19].