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Continuity of restrictions of f extends to continuity of f

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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.

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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.

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Georgia Martin, PhD
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  • AB, Hood College
  • PhD, The Catholic University of America
  • PhD, The University of Maryland at College Park
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