Monotone

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Let $(X_1,\tau_1)$ and $(X_2,\tau_2)$ be topological spaces. A function $f \colon X_1 \rightarrow X_2$ is called monotone if $f$ is continuous and if for every $y \in X_2$, $f^{-1}(y)$ is connected.

Properties

Onto monotone map of an interval is an arc