J. van Mill

S4.3 and hereditarily extremally disconnected spaces

Abstract The modal logic S4.3 defines the class of hereditarily extremally disconnected spaces (HED-spaces). We construct a countable HED-subspace X of the Gleason cover of the real closed unit interval [0,1] such that S4.3 is the logic of X.

Countable Dense Homogeneous Rimcompact Spaces and Local Connectivity

We prove that every nonmeager connected Countable Dense Homogeneous space is locally connected under some additional mild connectivity assumption. As a corollary we obtain that every Countable Dense Homogeneous connected rimcompact space is locally connected.

Superextensions which are Hilbert cubes

It is shown that each separable metric, not totally disconnected, topological space admits a superextension homeomorphic to the Hilbert cube. Moreover, for simple spaces, such as the closed unit interval or then-spheresSn, we give easily described subbases for which the corresponding superextension is homeomorphic to the Hilbert cube.

Compactifications of locally compact spaces with zero-dimensional remainder

Abstract For a locally compact space X we give a necessary and sufficient condition for every compactification aX of X with zero-dimensional remainder to be regular Wallman. As an application it follows that the Freudenthal compactification of a locally compact metrizable space is regular Wallman.

A New Proof of the McKinsey–Tarski Theorem

It is a landmark theorem of McKinsey and Tarski that if we interpret modal diamond as closure (and hence modal box as interior), then

On topological groups with a first-countable remainder, II

\mathsf S4

Nonhomogeneity of remainders

S4 is the logic of any dense-in-itself metrizable space. The McKinsey–Tarski Theorem relies heavily on a metric that gives rise to the topology. We give a new and more topological proof of the theorem, utilizing Bing’s Metrization Theorem.

Some aspects of dimension theory for topological groups

Abstract Recently, De Groots conjecture that cmp X = def X holds for every separable and metrizable space X has been negatively resolved by Pol. In previous efforts to resolve De Groots conjecture various functions like cmp have been introduced. A new inequality between two of these functions is established. Many examples which have been constructed so far in relation with the conjecture are obtained by attaching a locally compact space to a compact space. An upper bound for the compactness deficiency def of the resulting space is given.