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## Semicontinuity

General Topology
Tsakanikas Nickos
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### Semicontinuity

Definition: Let $Y$ be a topological space. A function $\varphi \ \colon Y \longrightarrow \mathbb{Z}$ is called upper semicontinuous if for every $y \in Y$ there exists an open neighborhood $U$ of $y$ such that $\varphi(y) \geq \varphi(y^{\prime})$ for all $y^{\prime} \in U$.

Show that a function $\varphi \ \colon Y \longrightarrow \mathbb{Z}$ is upper semicontinuous if and only if for every $n \in \mathbb{Z}$ the set $\left\{ \, y \in Y \ \big| \ \varphi(y) \geq n \, \right\}$ is a closed subset of $Y$.