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Published online by Cambridge University Press: 20 November 2018
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
$f\left( {{X}^{'}} \right)$ does not meet
$Z$? We also give a relative variant: if
$f\left( X\prime \right)$ 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.