The spaces of ordered lo-proximity
The span and the width of continua
The spectrum and commutant of a certain weighted translation operator.
The spectrum of a separated presheaf and the relation between totally fuzzy subsets and fuzzy subsets; application to fuzzy topology
The star compactification.
The strongest t-norm for fuzzy metric spaces
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.
The structure of atoms (hereditarily indecomposable continua)
Let X be an atom (= hereditarily indecomposable continuum). Define a metric ϱ on X by letting where is the (unique) minimal subcontinuum of X which contains x and y and W is a Whitney map on the set of subcontinua of X with W(X) = 1. We prove that ϱ is an ultrametric and the topology of (X,ϱ) is stronger than the original topology of X. The ϱ-closed balls C(x,r) = y ∈ X:ϱ ( x,y) ≤ r coincide with the subcontinua of X. (C(x,r) is the unique subcontinuum of X which contains x and has Whitney value...
The structure of compacta satisfying dim(X×X) < 2dimX
The structure of Eberlein, uniformly Eberlein and Talagrand compact spaces in Σ()
The structure of orbits in dynamical systems
The structure of the -ideal of -porous sets
We show a general method of construction of non--porous sets in complete metric spaces. This method enables us to answer several open questions. We prove that each non--porous Suslin subset of a topologically complete metric space contains a non--porous closed subset. We show also a sufficient condition, which gives that a certain system of compact sets contains a non--porous element. Namely, if we denote the space of all compact subsets of a compact metric space with the Vietoris topology...
The structure of -limit sets for continuous maps of the interval
We prove that every infinite nowhere dense compact subset of the interval is an -limit set of homoclinic type for a continuous function from to .
The Subset P+ And P- Of The Split Interval
The subsets P+ and P- of the split interval.
The subspace of weak -points of
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 .
The sup = max problem for the extent and the Lindelöf degree of generalized metric spaces, II
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...
The sup = max problem for the extent of generalized metric spaces
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.
The superextension of the closed unit interval is homeomorphic to the Hilbert cube
The superposition operator in Musielak-Orlicz spaces of vector-valued functions
The Suslinian number and other cardinal invariants of continua
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...