The Implicit Function Theorem and the Elementary Closure Theorem

Abstract

As in Section 2.15 let us consider briefly an abstract space S of elements x, and let us assume that a concept σ of convergence of sequences x _k of elements of S has been defined, satisfying the two main axioms: (a) If [x _k ] converges to x in S, then any subsequence $\left[ {{{x}_{{{{k}_{s}}}}}} \right]$ also converges to x; (b) Any sequence of repetitions [x, x,…,x,…] must converge to x, where x is any element of S. Any such space is called a σ-limit space. In Section 2.15 we introduced the concepts of σ-lower and σ-upper semicontinuity of a functional F:S → reals. A functional which is both upper and lower semicontinuous is said to be continuous. Let us show here that, already at this level of generality, quite relevant theorems can be proved. To this effect, let us carry over the usual concepts. Thus, we say that a subset A of S is σ-closed if all elements of accumulation of A in S belong to A; that is, if x₀ ∈ S is the σ-limit of elements x _k of A, then x₀ ∈ A. We say that a subset A of S is relatively sequentially σ-compact if every sequence [x _k ] of elements of A possesses a subsequence $\left[ {{{x}_{{{{k}_{s}}}}}} \right]$ which is σ-convergent to an element x of S.