Non-Hausdorff groupoids, proper actions and -theory.
It is known that every remainder of a topological group is Lindelöf or pseudocompact. Motivated by this result, we study in this paper when a topological group has a normal remainder. In a previous paper we showed that under mild conditions on , the Continuum Hypothesis implies that if the Čech-Stone remainder of is normal, then it is Lindelöf. Here we continue this line of investigation, mainly for the case of precompact groups. We show that no pseudocompact group, whose weight is uncountable...
J. Terasawa in " are non-normal for non-discrete spaces " (2007) and the author in “On non-normality points and metrizable crowded spaces” (2007), independently showed for any metrizable crowded space that each point of its Čech–Stone remainder is a non-normality point of . We introduce a new class of spaces, named nice spaces, which contains both of Sorgenfrey line and every metrizable crowded space. We obtain the result above for every nice space.
In this short article we answer the question posed in Ghadermazi M., Karamzadeh O.A.S., Namdari M., On the functionally countable subalgebra of , Rend. Sem. Mat. Univ. Padova 129 (2013), 47–69. It is shown that is isomorphic to some ring of continuous functions if and only if is functionally countable. For a strongly zero-dimensional space , this is equivalent to say that is functionally countable. Hence for every -space it is equivalent to pseudo--compactness.
Let be the Tychonoff product of -many Tychonoff non-single point spaces . Let be a point in the closure of some whose weak Lindelöf number is strictly less than the cofinality of . Then we show that is not normal. Under some additional assumptions, is a butterfly-point in . In particular, this is true if either or and is infinite and not countably cofinal.
We prove that for an unbounded metric space , the minimal character of a point of the Higson corona of is equal to if has asymptotically isolated balls and to otherwise. This implies that under a metric space of bounded geometry is coarsely equivalent to the Cantor macro-cube if and only if and . This contrasts with a result of Protasov saying that under CH the coronas of any two asymptotically zero-dimensional unbounded metric separable spaces are homeomorphic.