On the cardinality of separable pseudoradial spaces

Abstract

The aim of this paper is to consider questions concerning the possible maximum cardinality of various separable pseudoradial (in short: SP) spaces. The most intriguing question here is if there is in ZFC a regular (or just Hausdorff) SP space of cardinality $>\mathfrak{c}$. While this question is left open, we establish a number of non-trivial results that we list below.

• It is consistent with $MA+\mathfrak{c}={\mathrm{\aleph }}_2$ that there is a countably tight and compact SP space of cardinality $2^{\mathfrak{c}}$.

• If κ is a measurable cardinal then in the forcing extension obtained by adding κ many Cohen reals, every countably tight regular SP space has cardinality at most $\mathfrak{c}$.

• If $\kappa >{\mathrm{\aleph }}_1$ Cohen reals are added to a model of GCH then in the extension every pseudocompact SP space with a countable dense set of isolated points has cardinality at most $\mathfrak{c}$.

• If $\mathfrak{c}\le {\mathrm{\aleph }}_2$ then there is a 0-dimensional SP space with a countable dense set of isolated points that has cardinality greater than $\mathfrak{c}$.

Introduction

The class of pseudoradial spaces is a natural and well-studied class that is a generalization of the radial property in the same way that the class of sequential spaces is a generalization of the class of Frechet-Urysohn spaces. For a cardinal κ, a sequence $\{x_{\alpha }:\alpha <\kappa \}$ is said to converge to x in a space X, if for every neighborhood U of x there is a $\beta <\kappa$ such that $\{x_{\alpha }:\beta <\alpha <\kappa \}$ is a subset of U. A set A is radially closed in X if, for every cardinal $\kappa \le |X|$, no κ-sequence of points of A converges to a point not in A. A space X is pseudoradial if every radially closed set is closed. A set A is sequentially closed if no ω-sequence of points from A converges to a point not in A and a space is sequential if every sequentially closed set is closed.

As stated in the abstract, we are exploring the question of whether separable pseudoradial spaces of cardinality greater than $\mathfrak{c}$ exist. We may say that a space is large if it has cardinality greater than $\mathfrak{c}$. We thank A. Bella for informing us that it was shown in [2] that it is consistent that the usual product space $2^{{\omega }_1}$ is SP.

One of the most interesting results about compact pseudoradial spaces arose when Sapirovskii [19] proved that the continuum hypothesis implied that a compact space is pseudoradial so long as it is sequentially compact. A space is sequentially compact if every infinite sequence has a limit point. This was improved in [15] where it was shown that it follows from $\mathfrak{c}\le {\mathrm{\aleph }}_2$ that compact sequentially compact spaces are pseudoradial and that this bound can not be improved in ZFC. In this paper we are able to show that this same assumption of $\mathfrak{c}\le {\mathrm{\aleph }}_2$ is sufficient to produce examples of regular Hausdorff separable pseudoradial spaces of cardinality greater than $\mathfrak{c}$. Moreover, if $2^{{\mathrm{\aleph }}_1}={\mathrm{\aleph }}_2$, then this can be improved to having SP spaces of cardinality $2^{\mathfrak{c}}$. In this paper we will restrict our investigation to regular Hausdorff spaces and note that $2^{\mathfrak{c}}$ is the upper bound on the cardinality of any separable space.

Returning to the class of compact separable spaces, we recall that every pseudoradial compact space is sequentially compact. More generally in a pseudoradial space, a countable discrete set is closed if it contains no converging sequence, and so clearly, a compact pseudoradial space is sequentially compact. It is well-known that the sequential closure of a countable set has cardinality at most $\mathfrak{c}$ and therefore every sequential separable space has cardinality at most $\mathfrak{c}$. Balogh [3] proved that the proper forcing axiom implies that every compact space of countable tightness is sequential, and therefore compact separable spaces of countable tightness have cardinality at most $\mathfrak{c}$. Of course this paper of Balogh's was in answer to the celebrated Moore-Mrowka problem. Recall that the proper forcing axiom also implies that $\mathfrak{c}={\mathrm{\aleph }}_2$ [17], [21]. This background motivates one to ask more about separable pseudoradial spaces of countable tightness both with and without the extra assumption of compactness.