Epsilon map

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Let $\epsilon > 0$ be a positive real number. Let $(X,\tau)$ and $(Y,\sigma)$ be metric spaces. We say that a continuous function $f \colon X \rightarrow Y$ is an $\epsilon$-map if it has the property that for all $y \in Y$, the diameter of $f^{-1}(y)$ obeys $0<\mathrm{diam} f^{-1}(y)<\epsilon$.