If κ < λ are such that κ is both supercompact and indestructible under κ-directed closed forcing which is also (κ+,∞)-distributive and λ is 2λ supercompact, then by [3, Theorem 5], {δ < κ | δ is δ+ strongly compact yet δ isn’t δ+ supercompact} must be unbounded in κ. We show that the large cardinal hypothesis on λ is necessary by constructing a model containing a supercompact cardinal κ in whic...

If κ < λ are such that κ is a strong cardinal whose strongness is indestructible under κ-strategically closed forcing and λ is weakly compact, then we show that A = {δ < κ | δ is a non-weakly compact Mahlo cardinal which reflects stationary sets} must be unbounded in κ. This phenomenon, however, need not occur in a universe with relatively few large cardinals. In particular, we show how to cons...

Assume that κ and λ are respectively strong weakly compact cardinals with λ>κ. Fix Θ≥λ a cardinal cof(Θ)>κ cof(δ)=δ<κ. Assuming the GCH≥κ holds, we construct generic extension of universe where is limit cardinal, cof(κ)=δ, 2κ=Θ TP(κ++) holds. This extends main result [5] for uncountable cofinalities.

A separated cell A of a topological space X is a family of pairwise disjoint open sets with the property that, for any partition A= A1 ∪ A2 of A, ⋃ A1 and ⋃ A2 are completely separated. The separated cellularity sc(X) of X is the supremum of all cardinals of separated cells. A space X has finite separation if it has no infinite separated cells. With X compact, every separated cell of X has size...

For a cardinal κ and a model M of cardinality κ let No(M) denote the number of non-isomorphic models of cardinality κ which are L∞κ-equivalent to M. In [She82] Shelah established that when κ is a weakly compact cardinal and μ ≤ κ is a nonzero cardinal, there exists a model M of cardinality κ with No(M) = μ. We prove here that if κ is a weakly compact cardinal, the question of the possible value...

We construct a model containing a proper class of strongly compact cardinals in which no strongly compact cardinal κ is κ supercompact and in which every strongly compact cardinal has its strong compactness resurrectible.

We generalize ∇(A), which was introduced in [2], to larger cardinals. For a regular cardinal κ > א0 we denote by ∇κ(A) the statement that A ⊂ κ and for all regular θ > κ do we have that {X ∈ [Lθ[A]] <κ : X ∩ κ ∈ κ ∧ otp(X ∩ OR) ∈ Card} is stationary in [Lθ[A]] . It was shown in [2] that ∇א1(A) can hold in a set-generic extension of L. We here prove that ∇א2(A) can hold in a semi-proper set-gene...

in this paper, we introduce the concept of rarely generalized regular fuzzy continuous functions in the sense of a.p. sostak&apos;s and ramadan is introduced. some interesting properties and characterizations of them are investigated. also, some applications to fuzzy compact spaces are established.

A graph is singular if the zero eigenvalue is in the spectrum of its 0-1 adjacency matrix A. If an eigenvector belonging to the zero eigenspace of A has no zero entries, then the singular graph is said to be a core graph. A (κ, τ)-regular set is a subset of the vertices inducing a κ-regular subgraph such that every vertex not in the subset has τ neighbours in it. We consider the case when κ = τ...

We investigate the problem if every compact space K carrying a Radon measure of Maharam type κ can be continuously mapped onto the Tikhonov cube [0, 1] (κ being an uncountable cardinal). We show that for κ ≥ cf(κ) ≥ ω2 this holds if and only if κ is a precaliber of measure algebras. Assuming that there is a family of ω1 null sets in 21 such that every perfect set meets one of them, we construct...