On basic embeddings of compacta into the plane

Abstract

A compactum K ⊂ ℝ2 is said to be basically embedded in ℝ2 if for each continuous function f: K → ℝ there exist continuous functions g, h: ℝ → ℝ such that f(x, y) = g(x) + h(y) for each point (x, y) ∈ K. Sternfeld gave a topological characterization of compacta K which are basically embedded in ℝ2 which can be formulated in terms of special sequences of points called arrays, using arguments from functional analysis. In this paper we give a simple topological proof of the implication: if there exists an array in K of length n for any n ∈ ℕ, then K is not basically embedded.

A simpler proof of the Sternfeld theorem

“…, S − 1} we have a α,2−ρ 2 (α) = a α+1,2−ρ 2 (α) . The definition is a specific case of [MKT,§2,Definition 2]. See an example of a Sternfeld array in the plane in Figure 1.…”

Constructive decomposition of a function of two variables as a sum of functions of one variable

“…It is therefore desirable to find a straightforward, constructive proof which will consequently provide an elementary proof of Skopenkov's and Kurlin's characterizations. A constructive proof of (B) ⇒ (A) is given in [9]. In this paper we give an elementary construction proving the implication (A) ⇒ (B) provided that m = 2:…”

Constructive decomposition of a function of two variables as a sum of functions of one variable

“…It is therefore desirable to find a straightforward, constructive proof which will consequently provide an elementary proof of Skopenkov's and Kurlin's characterizations. A constructive proof of (B) ⇒ (A) is given in [MKT03].…”