### A Space Of Small Spread Without
The Usual Properties

Proceedings Of The
American Mathematical Society, Vol. 43, No. 1 (March 1974), 245-248

Received by the editors June 12, 1973.

ABSTRACT. A space is found, for any α, which has spread α and which is not the set-theoretic union of a hereditarily α-Lindelof and a hereditarily α-separable space.

Introduction

At the 1972 Bolyai Janos Mathematical Society Colloquium, A. Hajnal and I. Juhasz noted that every known Hausdorff space of spread ω was the union of a hereditary separable space and a hereditarily Lindelof space. The main result of this paper is a family of counterexamples to a generalization of this situation; the method of proof will also yield, in Lemma 2(c), a family of spaces such that no "large" subspaces are regular.