We prove that the existence of a measurable cardinal is equivalent to the existence of a normal space whose modal logic coincides with the modal logic of the Kripke frame isomorphic to the powerset of a two element set.

Urysohn’s lemma is a crucial property of normal spaces that deals with separation of closed sets by continuous functions. It is also a fundamental ingredient in proving the Tietze extension theorem, another property of normal spaces that deals with the existence of extensions of continuous functions. Using the Cantor function, we give alternative proofs for Urysohn’s lemma and the Tietze extension theorem.

Let A = C(X) ⊗ K(H), where X is a compact Hausdorff space and K(H) is the algebra of compact operators on a separable infinite-dimensional Hilbert space. Let A s be the algebra of strong*-continuous functions from X to K(H). Then A s /A is the inner corona algebra of A. We show that if X has no isolated points, then A s /A is an essential ideal of the corona algebra of A, and Prim(A s /A), the primitive ideal space of A s /A, is not weakly Lindelof. If X is also first countable, then there is a natural injection from the power set of X to the lattice of closed ideals of A s /A. If X = βℕ\ℕ and the continuum hypothesis (CH) is assumed, then the corona algebra of A is a proper subalgebra of the multiplier algebra of A s /A. Several of the results are obtained in the more general setting of C 0(X)-algebras.

We establish that if it is consistent that there is a supercompact cardinal, then it is consistent that every locally compact, hereditarily normal space that does not include a perfect pre-image of ${{\text{ }\!\!\omega\!\!\text{ }}_{1}}$ is hereditarily paracompact.

Extending the work of Larson and Todorcevic, we show that there is a model of set theory in which normal spaces are collectionwise Hausdorff if they are either first countable or locally compact, and yet there are no first countable $L$-spaces or compact $S$-spaces. The model is one of the form $\text{PFA}\left( S \right)\left[ S \right]$, where $S$ is a coherent Souslin tree.

In an $\text{H}$-closed, Urysohn space, disjoint $\text{H}$-sets can be separated by disjoint open sets. This is not true for an arbitrary H-closed space even if one of the $\text{H}$-sets is a point. In this paper, we provide a systematic study of those spaces in which disjoint $\text{H}$-sets can be separated by disjoint open sets.

A continuumis said to be Suslinian if it does not contain uncountably many mutually exclusive nondegenerate subcontinua. We prove that Suslinian continua are perfectly normal and rim-metrizable. Locally connected Suslinian continua have weight atmost ${{\omega }_{1}}$ and under appropriate set-theoretic conditions are metrizable. Non-separable locally connected Suslinian continua are rim-finite on some open set.

It is known that the product ${{\omega }_{1}}\times X$ of ${{\omega }_{1}}$ with an ${{M}_{1}}$-space may be non-normal. In this paper we prove that the product $\kappa \times X$ of an uncountable regular cardinal κ with a paracompact semi-stratifiable space is normal iff it is countably paracompact. We also give a sufficient condition under which the product of a normal space with a paracompact space is normal, from which many theorems involving such a product with a countably compact factor can be derived.

It is proved that for every Hausdorff space ℝ and for every Hausdorff (regular or Moore) space X, there exists a Hausdorff (regular or Moore, respectively) space S containing X as a closed subspace and having the following properties:

• la) Every continuous map of S into ℝ is constant.

• b) For every point x of S and every open neighbourhood U of x there exists an open neighbourhood V of x, V ⊆ U such that every continuous map of V into ℝ is constant.

• 2) Every continuous map f of S into S (f ≠ identity on S) is constant.

A topological space X is said to be somewhat normal provided that for each open cover is a normal cover of X. We show that a completely regular somewhat normal space need not be normal, thereby answering a question of W. M. Fleischman. We note that a collectionwise normal somewhat normal space need not be almost 2-fully normal, as had previously been asserted, and that neither the perfect image nor the perfect preimage of a somewhat normal space has to be somewhat normal.

We prove that a topological space X has a locally connected regular T1, extension if and only if X is the underlying topological space of a nearness space Y which is concrete, regular and uniformly locally uniformly connected.

In the mid 1970's, Shelah formulated a weak version of ◊. This axiom Φ is a prediction principle for colorings of the binary tree of height ω1. Shelah and Devlin showed that Φ is equivalent to 2ℵ0 < 2ℵ1.

In this paper, we formulate Φp, a "Φ for partial colorings", show that both ◊* and Fleissner's “◊ for stationary systems” imply Φp, that ◊ does not imply Φp and that Φp does not imply CH.

We show that Φp implies that, in a normal first countable space, a discrete family of points of cardinality ℵ1 is separated.

A short proof is given of an internal characterization of completely regular spaces due to J. Kerstan.

In this paper the notion of inverse cluster set, which was recently introduced and studied for functions by T. R. Hamlett and P. E. Long (Proc. Amer. Math. Soc, 53 (1975), 470-476), is extended to and investigated for multifunctions. We generalize the notion of inverse cluster set, extend to multifunctions and generalize some known results for inverse cluster sets of functions and offer some new results. In the latter sections, compactness generalizations are characterized in terms of inverse cluster sets and some results on connected and conectivity functions are extended to multifunctions.

It is proved, in particular, that a topological space X is a Baire space if and only if every real valued function f: X →R is almost continuous on a dense subset of X. In fact, in the above characterization of a Baire space, the range space R of real numbers may be generalized to any second countable, Hausdorfï space that contains infinitely many points.