On functions that approximate relations

Abstract:

Let $X$ be a metric space and let $Y$ be a separable metric space. Suppose $R$ is a relation in $X \times Y$. The following are equivalent: (a) for each $\varepsilon > 0$ there exists $f:X \to Y$ such that the Hausdorff distance from $f$ to $R$ is at most $\varepsilon$; (b) the domain of $R$ is a dense subset of $X$, and for each isolated point $x$ of the domain the vertical section of $R$ at $x$ is a singleton; (c) for each $\varepsilon > 0$ there exists $f:X \to Y$ of Baire class one such that the Hausdorff distance from $f$ to $R$ is at most $\varepsilon$.