Topological Properties of Subsets in Real Numbers 1

The notation and terminology used in this paper have been introduced in the following articles: [9], [3], [10], [1], [2], [7], [5], [6], [4], and [8]. For simplicity we adopt the following convention: n, m are natural numbers, x is arbitrary, s, g, g1, g2, r, p, q are real numbers, s1, s2 are sequences of real numbers, and X, Y , Y1 are subsets of . In this article we present several logical schemes. The scheme SeqChoice concerns a non-empty set A, and a binary predicate P, and states that: there exists a function f from into A such that for every element t of holds P[t, f(t)] provided the following requirement is met: • for every element t of there exists an element ff of A such that P[t, ff ]. The scheme RealSeqChoice concerns a binary predicate P, and states that: there exists s1 such that for every n holds P[n, s1(n)] provided the parameter meets the following requirement: • for every n there exists r such that P[n, r]. We now state several propositions: (1) X ⊆ Y if and only if for every r such that r ∈ X holds r ∈ Y . (2) r ∈ X if and only if r / ∈ X.