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    Box topology does not preserve first countable

    Let X_n=\mathbb{R} and X= \prod_{n \in \mathbb{N}} X_n = \mathbb{R}^{\mathbb{N}}. Suppose U_{n \in \mathbb{N}} is a countable neighborhood basis for the point \vec{0} = (0,0, \ldots ) \in X. See if you can think of an open neighborhood of \vec{0} that does not contain any one of the U_n's...
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