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Thomas plank is not realcompact
Let $X = \bigcup_{n\geq 0} L_n$ where $L_n = [0, 1)\times\{1/i\}$ for $i > 0$ and $L_0 = (0, 1)\times \{0\}$.
Define the topology on $X$ as follows: each point $(x, 1/i)$ for $x\in (0, 1)$ and $i &...
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