A note on O-uniformities and strongly proximal spaces

Let X   be a topological space, then a uniformity DD on X is called an O-uniformity on X   if for any x∈Xx∈X and for any D∈DD∈D the D  -ball D[x]D[x] about x is a neighborhood of x in X. A space X is called a subuniform space if there is an O-uniformity on X. We obtain some sufficient conditions that a uniform (subuniform) space is metrizable. A topological space X is metrizable if and only if there is some uniformity (O  -uniformity) DD on X   such that (X,D)(X,D) is a uniform (subuniform) space and there is a decreasing sequence {Dn:n∈N}{Dn:n∈N} of elements of DD such that if {xn}n∈N{xn}n∈N is a sequence of points in X   with y∈⋂n∈NDn[xn], then {xn}n∈N{xn}n∈N converges to y  . We introduce notions of weakly proximal and strongly proximal. We prove that if DD is an O-uniformity on a topological space X such that X   is weakly proximal with respect to DD then (X,D)(X,D) is proximal. Every strongly proximal space is perfect. We also get that every subspace of a strongly proximal space is strongly proximal, a countable product of strongly proximal spaces is strongly proximal.