Explore BrainMass

Explore BrainMass

    Continuity of restrictions of f extends to continuity of f

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    Let X refer to a general topological space.

    Suppose X = A1 ï?? A2 ï?? â?¦, where An ï? Ã...n+1 for each n. If f : X --> Y is a function such that, for each n, f |An : An --> Y is continuous with respect to the induced topology on An, show that f itself is continuous.

    Please note that Ã...n+1 (A with a small circle above it) denotes the interior of A "subscript" n+1 (That is, each set Ai is embedded in the interior of the subsequent set Ai+1).

    Also note that f | An : An --> Y denotes the restriction of the function x to the subset A "subsript" n.

    See attachment for better a better formatted question.

    © BrainMass Inc. brainmass.com October 10, 2019, 2:28 am ad1c9bdddf


    Solution Summary

    A detailed proof is provided of the fact that if X is the union of sets A_1, A_2, A_3, ... where, for every n, the set A_n is contained in the interior of the set A_{n+1} and the restriction of a function f to A_n is continuous, then f is continuous on X.