In the present paper, we introduce soft generalized closed sets in soft topological spaces which are defined over an initial universe with a fixed set of parameters. A sufficient condition for a soft g-closed set to be a soft closed is also presented. Moreover, the union and intersection of two soft g-closed sets are discussed. Finally, the new soft separation axiom, namely soft 1 2 T space is ...

The geometry conjecture, which was posed nearly a quarter of century ago, states that the fixed point set composition projectors onto nonempty closed convex sets in Hilbert space is actually equal to intersection certain translations underlying sets.

Using geodesic length functions, we define a natural family of real codimension 1 subvarieties of Teichmüller space, namely the subsets where the lengths of two distinct simple closed geodesics are of equal length. We investigate the point set topology of the union of all such hypersurfaces using elementary methods. Finally, this analysis is applied to investigate the nature of the Markoff conj...

The exterior sphere condition is compared to proximal smoothness, and examples are provided, which show that the two properties are not necessarily equivalent. Then conditions are given under which equivalence holds, and an open question involving the union of uniform closed balls property is stated in the form of a conjecture.

We prove that a real analytic subset of a torus group that is contained in its image under an expanding endomorphism is a finite union of translates of closed subgroups. This confirms the hyperplane zeros conjecture of Lagarias and Wang for real analytic varieties. Our proof uses real analytic geometry, topological dynamics, and Fourier analysis.

We give improved lower bounds on the size of Kakeya and Nikodym sets over Fq. We also propose a natural conjecture on the minimum number of points in the union of a not-too-flat set of lines in Fq, and show that this conjecture implies an optimal bound on the size of a Nikodym set. Finally, we study the notion of a weak Nikodym set and give improved, and in some special cases optimal, bounds fo...

In 1979, P. Frankl conjectured that in a finite union-closed family F of finite sets, F 6= {∅}, there has to be an element that belongs to at least half of the sets in F . We prove this when | ⋃

In 1979 Frankl conjectured that in a finite union-closed family F of finite sets, F = {∅} there has to be an element that belongs to at least half of the sets in F . We prove this when | F| 10.

We investigate attractors for compact sets by considering a certain quotient space. The following theorem is included. Let /: G -> G, G a closed convex subset of a Banach space, / a mapping satisfying (i) there exists M c G which is an attractor for compact sets under/; (ii) the family {/"}Ü°_i is equicontinuous. Then/has a fixed point. 1. Since the appearance of the celebrated Schauder fixed p...