On sequentially Ramsey sets
On some combinatorial problems involving large cardinals
On some geometric transformation of t-norms.
Given a triangular norm T, its t-reverse T*, introduced by C. Kimberling (Publ. Math. Debrecen 20, 21-39, 1973) under the name invert, is studied. The question under which conditions we have T** = T is completely solved. The t-reverses of ordinal sums of t-norms are investigated and a complete description of continuous, self-reverse t-norms is given, leading to a new characterization of the continuous t-norms T such that the function G(x,y) = x + y - T(x,y) is a t-conorm, a problem originally studied...
On some hypotheses concerning trees.
On Some Intercalations In Ordered Sets
On some isomorphisms of De Morgan algebras of fuzzy sets.
In this paper the classes of De Morgan algebras (P(X),∩,U,n) are studied. With respect to isomorphisms of such algebras, being P(X) the fuzzy sets on a universe X taking values in [0,1], U and ∩ the usual union and intersection given by max and min operations and n a proper complement.
On some kinds of fuzzy connected spaces
In this paper we introduce new results in fuzzy connected spaces. Among the results obtained we can mention the good extension of local connectedness. Also we prove that in a -fuzzy compact space the notions c-zero dimensional, strong c-zero dimensional and totally -disconnected are equivalent.
On some problem of A. Rosłanowski
We present a negative answer to problem 3.7(b) posed on page 193 of [2], where, in fact, A. Rosłanowski asked: Does every set of Lebesgue measure zero belong to some Mycielski ideal?
On some problems in combinatorial set theory.
On some problems in
On some properties of Hamel bases and their applications to Marczewski measurable functions
We introduce new properties of Hamel bases. We show that it is consistent with ZFC that such Hamel bases exist. Under the assumption that there exists a Hamel basis with one of these properties we construct a discontinuous and additive function that is Marczewski measurable. Moreover, we show that such a function can additionally have the intermediate value property (and even be an extendable function). Finally, we examine sums and limits of such functions.
On some properties of squares of Sierpiński sets
We investigate some geometrical properties of squares of special Sierpiński sets. In particular, we prove that (under CH) there exists a Sierpiński set S and a function p: S → S such that the images of the graph of this function under π'(⟨x,y⟩) = x - y and π''(⟨x,y⟩) = x + y are both Lusin sets.
On some properties of -planes of type-2 fuzzy sets
Some basic properties of -planes of type-2 fuzzy sets are investigated and discussed in connection with the similar properties of -cuts of type-1 fuzzy sets. It is known, that standard intersection and standard union of type-1 fuzzy sets (it means intersection and union under minimum t-norm and maximum t-conorm, respectively) are the only cutworthy operations for type-1 fuzzy sets. Recently, a similar property was declared to be true also for -planes of type-2 fuzzy sets in a few papers. Thus,...
On somewhat fuzzy semicontinuous functions
In this paper the concept of somewhat fuzzy semicontinuous functions, somewhat fuzzy semiopen functions are introduced and studied. Besides giving characterizations of these functions, several interesting properties of these functions are also given. More examples are given to illustrate the concepts introduced in this paper.
On spaces with the ideal convergence property
Let I ⊆ P(ω) be an ideal. We continue our investigation of the class of spaces with the I-ideal convergence property, denoted (I). We show that if I is an analytic, non-countably generated P-ideal then (I) ⊆ s₀. If in addition I is non-pathological and not isomorphic to , then (I) spaces have measure zero. We also present a characterization of the (I) spaces using clopen covers.
On special partitions of Dedekind- and Russell-sets
A Russell set is a set which can be written as the union of a countable pairwise disjoint set of pairs no infinite subset of which has a choice function and a Russell cardinal is the cardinal number of a Russell set. We show that if a Russell cardinal has a ternary partition (see Section 1, Definition 2) then the Russell cardinal fails to have such a partition. In fact, we prove that if a ZF-model contains a Russell set, then it contains Russell sets with ternary partitions as well as Russell...
On splitting infinite-fold covers
Let X be a set, κ be a cardinal number and let ℋ be a family of subsets of X which covers each x ∈ X at least κ-fold. What assumptions can ensure that ℋ can be decomposed into κ many disjoint subcovers? We examine this problem under various assumptions on the set X and on the cover ℋ: among other situations, we consider covers of topological spaces by closed sets, interval covers of linearly ordered sets and covers of ℝⁿ by polyhedra and by arbitrary convex sets. We focus on...
On squares, outside guessing of clubs and I
On star covering properties related to countable compactness and pseudocompactness
We prove a number of results on star covering properties which may be regarded as either generalizations or specializations of topological properties related to the ones mentioned in the title of the paper. For instance, we give a new, entirely combinatorial proof of the fact that -spaces constructed from infinite almost disjoint families are not star-compact. By going a little further we conclude that if is a star-compact space within a certain class, then is neither first countable nor separable....
On strong measure zero subsets of
We study the generalized Cantor space and the generalized Baire space as analogues of the classical Cantor and Baire spaces. We equip with the topology where a basic neighborhood of a point η is the set ν: (∀j < i)(ν(j) = η(j)), where i < κ. We define the concept of a strong measure zero set of . We prove for successor that the ideal of strong measure zero sets of is -additive, where is the size of the smallest unbounded family in , and that the generalized Borel conjecture...