The Definition of Topological Manifolds

Let x, y be sets. Observe that {〈x, y〉} is one-to-one. In the sequel n denotes a natural number. One can prove the following two propositions: (1) For every non empty topological space T holds T and T ΩT are homeomorphic. (2) Let X be a non empty subspace of En T and f be a function from X into R1. Suppose f is continuous. Then there exists a function g from X into En T such that (i) for every point a of X and for every point b of En T and for every real number r such that a = b and f(a) = r holds g(b) = r · b, and (ii) g is continuous. Let us consider n and let S be a subset of En T. We say that S is ball if and only if: (Def. 1) There exists a point p of En T and there exists a real number r such that S = Ball(p, r).