On imbedding of closed 2-dimensional disks into $R^2$

Abstract

Let $X$ be a topological space, $U$ -- opened subset of $X$. We will say that point $x \in \partial U$ is {\it accessible} from $U$ if there exists continuous injective mapping $ϕ: I \to \Cl D$ such that $ϕ(1)=x$, $ϕ([0,1)) \subset \Int U$. We proove the next main theorem. The following conditions are neccesary and suffficient for a compact subset $D$ of $R^2$ with a nonempty interior $\Int D$ to be homeomorphic to a closed 2-dimensional disk: 1) sets $\Int D$ and $R^2 \setminus D$ are connected; 2) any $x \in \partial D$ is accessible both from $\Int D$ and from $R^2 \setminus D$.