According to a result of one of the authors [7] there are at most a countable number of inequivalent differentiable actions of a compact Lie group on a compact differentiable manifold. The results above show that both the compactness of the manifold and the differentiability of the action are necessary assumptions. In the course of our proof we also prove the following theorem, which is an elem...

We introduce the class of countably I-compact spaces as a proper subclass of countably S-closed spaces. A topological space (X,T) is called countably I-compact if every countable cover of X by regular closed subsets contains a finite subfamily whose interiors cover X. It is shown that a space is countably I-compact if and only if it is extremally disconnected and countably S-closed. Other chara...

In this paper we study an example-machine Bush(S, T ) where S and T are disjoint dense subsets of R. We find some topological properties that Bush(S, T ) always has, others that it never has, and still others that Bush(S, T ) might or might not have, depending upon the choice of the disjoint dense sets S and T . For example, we show that every Bush(S, T ) has a point-countable base, is heredita...

In this paper, we generalize Vaughan’s and Bonanzinga’s results on absolute countable compactness of product spaces and give an example of a separable, countably compact, topological group which is not absolutely countably compact. The example answers questions of Matveev [8, Question 1] and Vaughan [9, Question (1)].

The concepts of P-compactness, countable P-compactness, the P-Lindelöf property are introduced in L-topological spaces by means of preopen L-sets and their inequalities when L is a complete DeMorgan algebra. These definitions do not rely on the structure of the basis lattice L and no distributivity in L is required. They can also be characterized by means of preclosed L-sets and their inequalit...

This is a continuation of [dhhkr]. We study the Tychonoff Compactness Theorem for various definitions of compact and for various types of spaces, (first and second countable spaces, Hausdorff spaces, and subspaces of R). We also study well ordered Tychonoff products and the effect that the multiple choice axiom has on such products.

Conditions on compact convergences to be topological are (and have been) investigated. New variants are obtained for both filter and cover compact-like properties such as countable compactness and Lindelöfness. The recently developed coreflectively modified continuous duality plays a crucial role in this quest. In particular, the results on compactness evoked above can be obtained either as byp...

We prove that the Continuum Hypothesis is consistent with the statement that countably compact regular spaces that are hereditarily of countable π–character are either compact or contain an uncountable free sequence. As a corollary we solve a well–known open question by showing that the existence of a compact S–space of size greater than א1 does not follow from the Continuum Hypothesis.

We de*ne the clique-width of a countable graph. We prove that a countable graph has *nite clique-width i+ its *nite induced subgraphs have bounded clique-width. We obtain an application to a conjecture concerning the structure of sets of countable graphs having a decidable monadic second-order satis*ability problem. c © 2003 Elsevier B.V. All rights reserved.