Two compact spaces are co-absoluteif their respective regular open algebras are isomorphic (i.e. homeomorphic Gleason covers). We prove that it is consistent that βω and βℝ are not co-absolute.
The relativization of Gryzlov’s theorem about the size of compact -spaces with countable pseudocharacter is false.
In this paper, we prove that for a given positive continuous t-norm there is a fuzzy metric space in the sense of George and Veeramani, for which the given t-norm is the strongest one. For the opposite problem, we obtain that there is a fuzzy metric space for which there is no strongest t-norm. As an application of the main results, it is shown that there are infinite non-isometric fuzzy metrics on an infinite set.
Let be the subspace of consisting of all weak -points. It is not hard to see that is a pseudocompact space. In this paper we shall prove that this space has stronger pseudocompact properties. Indeed, it is shown that is a -pseudocompact space for all .
In [The sup = max problem for the extent of generalized metric spaces, Comment. Math. Univ. Carolin. The special issue devoted to Čech 54 (2013), no. 2, 245–257], the author and Yajima discussed the sup = max problem for the extent and the Lindelöf degree of generalized metric spaces: (strict) -spaces, (strong) -spaces and semi-stratifiable spaces. In this paper, the sup = max problem for the Lindelöf degree of spaces having -diagonals and for the extent of spaces having point-countable bases...
It looks not useful to study the sup = max problem for extent, because there are simple examples refuting the condition. On the other hand, the sup = max problem for Lindelöf degree does not occur at a glance, because Lindelöf degree is usually defined by not supremum but minimum. Nevertheless, in this paper, we discuss the sup = max problem for the extent of generalized metric spaces by combining the sup = max problem for the Lindelöf degree of these spaces.
By the Suslinian number Sln(X) of a continuum X we understand the smallest cardinal number κ such that X contains no disjoint family ℂ of non-degenerate subcontinua of size |ℂ| > κ. For a compact space X, Sln(X) is the smallest Suslinian number of a continuum which contains a homeomorphic copy of X. Our principal result asserts that each compact space X has weight ≤ Sln(X)⁺ and is the limit of an inverse well-ordered spectrum of length ≤ Sln(X)⁺, consisting of compacta with weight ≤ Sln(X) and...
We construct from ⋄ a T₂ example of a hereditarily Lindelöf space X that is not a D-space but is the union of two subspaces both of which are D-spaces. This answers a question of Arhangel'skii.