Some properties of semi-linear uniform spaces

Abstract

Semi-linear uniform space is a new space defined by Tallafha, A and Khalil, R in [3], the authors studied some cases of best approximation in such spaces, and gave some open problems in approximation theory in uniform spaces. Besides they defined a set valued map ρ on X × X and asked two questions about the properties of ρ. The purpose of this paper is to answer these questions. Besides we shall define another set valued map δ on X × X and give more properties of semi-linear uniform spaces using the maps ρ and δ. Also we shall give an example of a semi-linear uniform space which is not metrizable.

Fixed Point in Semi-linear Uniform Spaces and Convex Metric Spaces

“…In [12], [13] and [14] Tallafha, A. answered the question "Is there a semilinear uniform space which is not metrizable?". Also he defined another set valued map called δ on X × X, which is used with ρ to give more properties of semi-linear uniform spaces.…”

“…In [12],Tallafha gave some important properties of semi-linear uniform spaces, using the set valued map ρ and δ, some of these properties are given in the following proposition. Proposition 1.6.…”

“…Proposition 1.6. [12]Let (X, Γ) be a semi-linear uniform space. Then, i) If V ∈ Γ c (x,y) , then V ⫋ ρ(x, y) .…”

More Properties of Semi-Linear Uniform Spaces

“…x y δ In [11], Tallafha gave some important properties of semi-linear uniform spaces, using the set valued map ρ and δ . Now we shall give more properties of semi-linear uniform spaces.…”