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I'm trying to figure out if the following set is open or closed.

Let $X$ be a topological space and let $\bigcup_i U_i $ be an open covering. Suppose $V$ is a closed subspace of $X$. Then is $ U_i \cap V$ open or closed in $V$ ?

All I know is that $U_i \cap V$ is closed in $U_i$ but I can't see how this helps.

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    How do you define "being an open set in $V$ [for the induced topology]"? This may give you some insight. – 2017-01-30
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    I guess we would use the subspace topology? So A is open in X iff $A\cap V $ is open in V? Or have got that the wrong way around? – 2017-01-30
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    $A$ is open in $V$ iff there exists $O$ open in $X$ such that $A=O\cap V$. – 2017-01-30
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    Ah ok, so it's pretty much by definition then? – 2017-01-30
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    Yes -- each $U_i$ is open. – 2017-01-30

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