We prove several theorems connecting infinite series of real terms in relation to Borel and Baire classification of sets and functions, connectedness of mappings, porosity character of sets etc. | Turk J Math 26 (2002) , 339 – 353. ¨ ITAK ˙ c TUB On Some Properties Connecting Infinite Series B. K. Lahiri and P. Das Abstract We prove several theorems connecting infinite series of real terms in relation to Borel and Baire classification of sets and functions, connectedness of mappings, porosity character of sets etc. Key words and phrases: Borel sets, connectedness, Baire classification of functions, porosity of sets. 1. Introduction The motivation for this paper arises from those of ([6], [7]; see also [2], [5]), where in [6], ∞ X an xn , an ’s [7] the authors proved several interesting and deep theorems on power series n=0 and x are real, in relation to Borel classification of sets, Baire classification of functions, connectedness of mappings, porosity character and Hausdorff dimension of sets etc. Our approach in this paper is somewhat different. Instead of power series, we consider infinite series of real terms and after defining a mapping suitably, we mainly study the behaviour of the mapping from various aspects. Since our context is different from [6], [7], the technique of proofs of the theorems also differ considerably from [6],[7]. 1991 AMS Subject Classification: 40A05. 339 LAHIRI, DAS 2. Definitions We consider the set s of all real sequences a = {ak } with the Fr´echet metric d(a, b) given by d(a, b) = ∞ X 1 |ak − bk | , 2k 1 + |ak − bk | k=1 where a = {ak } ∈ s and b = {bk } ∈ s. It is known that the metric space (s, d) is complete and has the power of the continuum. We introduce a mapping σ on the set s in the following way. If a = {ak } ∈ s, then σ(a) = lim sup Sn where Sn is the nth partial sum of the series Σak . Sn = n→∞ a1 +a2 +. . .+an . Clearly σ is a mapping from s to [− ∝, ∝]. In particular if a = {ak } ∈ s be such that Σak is convergent or properly divergent then σ(a) = Σak . If x, y etc. are members of s, we shall represent them generally by x = {xk }, y = {yk } etc. Also N denotes the set of positive integers .
