A topological space is said to be -separable if has a -closed-discrete dense subset. Recently, G. Gruenhage and D. Lutzer showed that -separable PIGO spaces are perfect and asked if -separable monotonically normal spaces are perfect in general. The main purpose of this article is to provide examples of -separable monotonically normal spaces which are not perfect. Extremely normal -separable spaces are shown to be stratifiable.
We show that there exists an Abelian topological group such that the operations in cannot be extended to the Dieudonné completion of the space in such a way that becomes a topological subgroup of the topological group . This provides a complete answer to a question of V.G. Pestov and M.G. Tkačenko, dating back to 1985. We also identify new large classes of topological groups for which such an extension is possible. The technique developed also allows to find many new solutions to the...
Assuming OCA, we shall prove that for some pairs of Fréchet -spaces , the Fréchetness of the product implies that is . Assuming MA, we shall construct a pair of spaces satisfying the assumptions of the theorem.
We investigate the fixed point property for tree-like continua that are unions of tree-like continua. We obtain a positive result if finitely many tree-like continua with the fixed point property have dendrites for pairwise intersections. Using Bellamy's seminal example, we define (i) a countable wedge X̂ of tree-like continua, each having the fpp, and X̂ admitting a fixed-point-free homeomorphism, and (ii) two tree-like continua H and K such that H, K, and H∩ K have the fixed point property, but...
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.
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 present an example of a connected, Polish, countable dense homogeneous space X that is not strongly locally homogeneous. In fact, a nontrivial homeomorphism of X is the identity on no nonempty open subset of X.
We show that is not normal, if is a limit point of some countable subset of , consisting of points of character . Moreover, such a point is a Kunen point and a super Kunen point.
A Mazurkiewicz set is a subset of a plane with the property that each straight line intersects in exactly two points. We modify the original construction to obtain a Mazurkiewicz set which does not contain vertices of an equilateral triangle or a square. This answers some questions by L.D. Loveland and S.M. Loveland. We also use similar methods to construct a bounded noncompact, nonconnected generalized Mazurkiewicz set.