Singular Discs Research Articles

Approximation of certain continuous functions of 𝑆² into 𝐸³

Bing has shown [I] that a homeomorphism of S2 into E3 may be polyhedrally approximated as close as desired. In this paper, we present sufficient conditions to assure that continuous functions of S2 into E3 may be approximated by homeomorphisms. The usual metric for S2 and E3 will be denoted by p. A disk is the homeomorphic image of [0, 1 ] X [0, 1 ] = 12. A singular disk is the continuous image of 12 which cannot be realized by a homeomorphism. If A is a subset of a singular disk D and f is the associated function from 12 to D, A will be called nonsingular if f1 restricted to A is a function. If f and g are continuous functions from X into Y, the distarnce from f to g, p(f, g), is the least upper bound of the set of all numbers p(f(x), g(x)) for xCX; d(f) is the greatest lower bound of the set of all numbers p(f, h) where h is a homeomorphism of X into Y and indicates how closely f may be approximated by a homeomorphism. If f is a continuous function from X into Y, the set of singular points of f, K(f), is the closure of the set of all x EX with the property that there is y EX, y x, such that f(y) =f(x). The following will be proven: