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Semicontinuity

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Tsakanikas Nickos
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Semicontinuity

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Post by Tsakanikas Nickos » Sat Jul 30, 2016 11:27 am

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 $Y$.
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