Between Arzelà and Whitney convergence. (English) Zbl 1071.54001

Ewert, Janina; Jędrzejewski, Jacek

Real Anal. Exch. 29(2003-2004), No. 1, 257-264 (2004).

The paper is devoted to convergences of nets of functions. A new convergence (Arzelà-Whitney) is defined and its relationships to various known convergences (Whitney, uniform, Arzelà, local Arzelà, pointwise) are studied. Let \(X\) be a topological space (no separation axioms are assumed) and let \(Y\) be a metric space. Let \(\{f_j:j\in J\}\) be a net of functions on \(X\) into \(Y\) and let \(f\) be a function on \(X\) into \(Y\). If the net converges to \(f\) pointwise and for every positive real continuous function \(\varphi\) and every \(j_0\in J\) there exists a finite subset \(J_1\) of \(J\) such that \(j\geq j_0\) for all \(j\in J_1\) and \(\min\{\text{dist}(f_j(x),f (x)): j\in J_1\}<\varphi(x)\), for all \(x\in X\), then the net is said to be convergent to \(f\) in the sense of Arzelà-Whitney. Sample result:
Theorem 1. Let \(X\) be an almost compact space. If a net \(\{f_j: j\in J\}\) of continuous functions \(f_j: X\to Y\) is pointwise convergent to a continuous function \(f: X\to Y\), then this net is AW-convergent to the function \(f\).
The results imply some interesting corollaries, e.g.:
Corollary 1. For a topological space \(X\) the following conditions are equivalent:
1. \(X\) is pseudocompact;
2. Every sequence \((f_n)^\infty_{n= 1}\), where \(f_n\in C(X, \mathbb{R})\) which is pointwise convergent to a continuous function \(f:X\to \mathbb{R}\) is also AW-convergent to the function \(f\).
Corollary 2. Every almost compact space is pseudocompact.

Reviewer: Roman Frič (Košice)

MSC:

54A20

Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)

Keywords:

Arzelà convergence; Whitney convergence; AW-convergence; paracompact space; pseudocompact space; nets of functions