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# On Perturbations of Continuous Maps

Published:2011-08-15
Printed: Mar 2013
• Benoît Jacob,
University of Toronto, Dept. of Mathematics, Toronto, ON M5S 2E4
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## Abstract

We give sufficient conditions for the following problem: given a topological space $X$, a metric space $Y$, a subspace $Z$ of $Y$, and a continuous map $f$ from $X$ to $Y$, is it possible, by applying to $f$ an arbitrarily small perturbation, to ensure that $f(X)$ does not meet $Z$? We also give a relative variant: if $f(X')$ does not meet $Z$ for a certain subset $X'\subset X$, then we may keep $f$ unchanged on $X'$. We also develop a variant for continuous sections of fibrations and discuss some applications to matrix perturbation theory.
 Keywords: perturbation theory, general topology, applications to operator algebras / matrix perturbation theory