On imbedding of closed 2-dimensional disks into R²

Original · EN

Let X be a topological space, U -- opened subset of X. We will say that point x ∈ ∂ U is accessible from U if there exists continuous injective mapping ϕ: I → D such that ϕ(1)=x, ϕ([0,1)) ⊂ U. We proove the next main theorem. The following conditions are neccesary and suffficient for a compact subset D of R² with a nonempty interior D to be homeomorphic to a closed 2-dimensional disk: 1) sets D and R² D are connected; 2) any x ∈ ∂ D is accessible both from D and from R² D.