Prove that any strictly monotone sequence (Uα)α<γ of open subsets of X has countable length, i.e. γ is countable. Hint: Use the same idea as in the proof of (a). (c) Show that every monotone sequence (Uα)α<ω1 open subsets of X eventually stabilizes, i.e. there is γ < ω1 such that for all α < ω1 with α ≥ γ, we have Uα = Uγ. Hint: Use the regularity of ω1. (d) Conclude that parts (a), (b) and (c)...

Let A be a subset of a ring with cardinality |A| = N . The sum set and the product set are 2A = A+A = {a1 +a2 | ai ∈ A} and A = AA = {a1a2 | ai ∈ A}. The study of sum-product sets is to estimate the sizes of 2A and A asymptotically when N is very large. Since Erdős and Szemerédi conjectured that if A ⊂ Z, at least one of |2A| and |A2| should be bigger than cN for any constant c, a lot of work h...

In this paper a new class of sets called  regular generalized  $delta$-closed set (briefly rg$delta$-closed set)is introduced and its properties are studied. Several examples are provided to illustrate the behaviour of these new class of sets.

A necessary and sufficient condition in terms of lower cut sets are given for the insertion of a $gamma-$continuous function between two comparable real-valued functions.

in this work, we define a fuzzy soft set theory and its related properties. we then define fuzzy soft aggregation operator that allows constructing more efficient decision making method. finally, we give an example which shows that the method can be successfully applied to many problems that contain uncertainties.

The bijective connection graph encompasses a family of cube-based topologies, and $n$ -dimensional graphs include the hypercube almost all its variants with order notation="LaTeX">$2^{n}$ degree . Hence, it is important to design implement algorithms that work in grap...