# Closed subset of a Hilbert space

$Y_0\subset Y$ is a closed Hilbert subspace of $Y$ with finite codimension and a subspace $Y_1$ satisfies $Y_0 \subset Y_1 \subset Y$. Is $Y_1$ also closed?

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Consider $\pi(Y_1) \subset Y/Y_{0}$, noting that a set is closed in the quotient topology precisely when its inverse is closed in the original space.

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