Decomposition of Topologies Which Characterize the Upper and Lower Semicontinuous Limits of Functions

and Applied Analysis 3 The quasi-uniform space X,U will always be considered to be a topological space with the topology obtained by using as the family of all neighborhoods of a point x0 ∈ X all sets of the formU x0 {x ∈ X : x0, x ∈ U}, whereU runs overU. Such a topology is called topology of the quasi-uniformity. Note that each topology onX is induced by a quasi-uniformity see 8, 9 , the most familiar of which is the Pervin quasi-uniformity 8 . A quasi-uniform space X,U is said to be a uniform space, and the family U will be called a uniformity for X, if and only if the quasi-uniformity satisfies the symmetric relation: if U ∈ U, then U−1 ∈ U. For a quasi-uniformity U the smallest uniformity containing U has a base all sets of the form ̂ U ∩ ̂ U−1 where ̂ U runs over a prescribed base for U. Moreover, a topology is induced by a uniformity if and only if the space is completely regular 7, 10 . Let us familiarize the reader with the notion of statistical convergence, that first appeared in 1935 under the name of almost convergence in the celebrated monograph of Zygmund 11 . The definition of statistical convergence for sequences of real numbers was given by Fast in 12 and is based on the notion of asymptotic density of a subset of natural numbers. Let A ⊂ N and n ∈ N. Put A n : {k ∈ A : k ≤ n}. Then one defines ∂ A : lim inf n→∞ |A n | n , ∂ A : lim sup n→∞ |A n | n , 2.1 as the lower and upper asymptotic density of A, respectively. If ∂ A ∂ A , then ∂ A lim n→∞ |A n | n 2.2 is the asymptotic or natural density of A. All the three densities, if they exist, are in 0, 1 . We recall also that ∂ N \A 1− ∂ A forA ⊂ N. A setA ⊂ X is said to be statistically dense if ∂ A 1. Let us mention that the union and intersection of two statistically dense sets in N are also statistically dense. For additional properties of the asymptotic density, in a more general setting, the reader might consult 13 . A sequence xn n∈N in a topological space X is said to converge statistically or shortly, st-converge to x ∈ X, if for every neighborhood U of x, ∂ {n ∈ N : xn / ∈ U} 0. This will be denoted by xn n∈N st-τ → x, where τ is a topology on X. It was shown 14 see 15, 16 for X R that for first countable spaces this definition is equivalent to the statement: there exists a subset A of N with ∂ A 1 such that the sequence xn n∈A converges to x. Recently in 17 , Çakalli and Khan pointed out that the first countability is not a necessary condition. 3. Decomposition of Pointwise Convergence and Weakly Exhaustiveness in R Given F ∈ F0 X and > 0, a base for the standard uniformity for the topology of pointwise convergence τp on R consists of all entourages of the form F; p : {( f, g ) : ∀x ∈ F ∣∣f x − g x ∣∣ < }. 3.1 4 Abstract and Applied Analysis In what follows, we offer a decomposition of the pointwise convergence in upper and lower part to better visualize their hidden and opposite roles with respect to the lower and upper semicontinuity of limits and functions. Definition 3.1. Let X, d be metric space, F ∈ F0 X , and > 0. Consider the quasi-uniformity on R having as a base all sets of the form