Metric spaces

  • Thread starter mansi
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Here’s a problem I’ve been struggling with, for a while….
If (X,d) is a metric space and f:X-->X is a continuous function, then show that A={ x in X : f(x)=x} is a closed set.
One possible way that I can think of is defining a new function g(x) = f(x)-x .Then A={x in X : g(x) =0}. Now {0} is closed, so by property of continuity, inverse image of a closed set is closed. Thus A is closed.
However, I’d like to do this problem using the basic definition of closed sets, i.e. show that the complement of A is open….so if I pick up an element ‘p’ not belonging to A, I must find a radius r so that B(p,r) is contained in the complement of A. This is where I’m getting stuck… please help !!! :confused:
 

Answers and Replies

  • #2
AKG
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Use the definition of continuity. If f is continuous at p, then there is an open ball of radius r around p such that for all x in that ball, f(x) is arbitrarily close to f(p). Make that arbitrary distance, [itex]\epsilon[/itex] < |f(p)|.
 

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