In this paper, L-convex ?*-remotehood system is introduced and some characterizations of both L-convexity remotehood are obtained. Further, ?*-remote mapping presented its properties investigated. Based on this, quasi-uniformity quasiuniformity preserving introduced. It proved that quasi-uniform space Lconvex mutually induced. addition, the category spaces can be embedded into spaces.

In this paper, we characterize each of various forms T0, T1, T2, and pre-Hausdorff extended pseudo-quasi-semi metric spaces as well examine how these generalizations are related. Moreover, give some invariance properties T2 investigate the relationship among irreducible Ti,i = 1,2 spaces. Finally, present Urysohn?s Lemma Tietze Extention Theorem for

In this paper, the Urysohn, completely Hausdorff and completely regular axioms in $L$-topological spaces are generalized to $L$-fuzzy topological spaces. Each $L$-fuzzy topological space can be regarded to be Urysohn, completely Hausdorff and completely regular tosome degree. Some properties of them are investigated. The relations among them and $T_2$ in $L$-fuzzy topological spaces are discussed.

Atsuji has internally characterized those metric spaces X for which each real-valued continuous function on X is uniformly continuous as follows: (1) the set X' of limit points of X is compact, and (2) for each £ > 0, the set of points in X whose distance from X' exceeds e is uniformly discrete. We obtain these new characterizations: (a) for each metric space V, the Hausdorff metric on C(X, Y),...

The reference for definitions both in the above theorem and in what follows is Topological dynamics by W. H. Gottschalk and G. A. Hedlund, Providence, 1955. It is tacitly assumed both in the statement of the theorem and in the remainder of the paper that the uniformity of X induces the (Hausdorff) topology. We observe that it is obvious that (II) implies (I); the remainder of the paper is devot...

We introduce a coarse combinatorial description of the Weil-Petersson distance dWP(X, Y ) between two finite area hyperbolic Riemann surfaces X and Y . The combinatorics reveal a connection between Riemann surfaces and hyperbolic 3-manifolds conjectured by Thurston: the volume of the convex core of the quasi-Fuchsian manifold Q(X, Y ) with X and Y in its boundary is comparable to the Weil-Peter...

A systematic analysis is made of the character of the free and free abelian topological groups on uniform spaces and on topological spaces. In the case of the free abelian topological group on a uniform space, expressions are given for the character in terms of simple cardinal invariants of the family of uniformly continuous pseudometrics of the given uniform space and of the uniformity itself....

We review recent results which relate spectral theory of discrete one-dimensional Schrödinger operators over strictly ergodic systems to uniform existence of the Lyapunov exponent. In combination with suitable ergodic theorems this allows one to establish Cantor spectrum of Lebesgue measure zero for a large class of quasicrystal Schrödinger operators. The results can also be used to study non-u...