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

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