The Union of two compact sets in $\mathbb{R} ^2$ with connected complement has a connected complement

TL;DR

This paper provides a rigorous proof that the union of two compact sets in the plane with connected complements also has a connected complement, clarifying an assumption used in previous works.

Contribution

It offers a formal proof of a topological property regarding compact sets in the plane, filling a gap in prior literature.

Findings

Confirmed the connectedness of the complement of the union of two compact sets with connected complements

Clarified topological properties of the plane relevant to complex analysis and topology

Provided a rigorous foundation for previous assumptions used in universal Taylor series studies

Abstract

In the papers from Chui and Parnes (1971) and Luh (1972), as well on the paper from V.Nestoridis (1996) on the Universal Taylor series, it is used, without proof, that the union of two compact sets in $\mathbb{R} ^2$ with connected complement has a connected complement. In this work we present a rigorous proof of this fact, by studying some important topological properties of the plane.