Uniqueness of the open cone neighborhood

Abstract

Introduction.The space ^4X[0, °o) with ^4X0 identified to a point v is called the open cone OC(A) over A and the point v is called the vertex of the cone.Let A be a space.A point xEX is said to have an open cone neighborhood U if there is a homeomorphism / of some OC(^4) onto the open set U of A with/(i>) =x.Our first theorem is the following. Theorem1. Let U and V be any two open cone neighborhoods of a point x in a locally compact Hausdorff space X.Then there is a homeomorphism of V onto U which leaves a neighborhood of x pointwise fixed.