Talk:Topology/Continuity and Homeomorphisms

The definition says, a function f from X to Y (topological spaces) is continuous if for all open sets U in Y the inverse image of U is an open set in X. This is not right because the inverse image of an element in Y is a SET of elements in X, in general. That is, its a set of sets. The inverse image of U isn't in X at all. Since I never took a class on topology I don't know how this definition should go... perhaps EACH OF THE ELEMENTS IN the inverse image should be open sets in X? 70.177.40.139 02:01, 14 July 2006 (UTC)

The definition is correct. You are also correct: the inverse of an element is indeed a set. The notation $$f^{-1}$$ simply means $$f^{-1}(U) = \{x \in X: f(x) \in U\}$$. See continuity on metric spaces. UvGroovy 10:08, 23 December 2006 (UTC)

Credit
As the original definition for homeomorphism was unclear, I have imported the current one from Wikipedia

SPat (talk) 08:39, 2 April 2008 (UTC)