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Due October 2nd

Problem 1

For any subset $A$ of the metric space $X$, $\diam A = \diam \bar A$.

Problem 2

Let $A\subset X$ be dense in $X$. If $E$ is closed in $X$ and $E\cap A = \emptyset$, then $E$ is nowhere dense.

Problem 3

If $A\subset X$ is a $G_\delta$ set and dense in $X$, then $X\setminus A$ is of the first category.

Problem 4

If $A$ and $X\setminus A$ are dense in the complete space $X$, then only one of them can be $F_\sigma$ in $X$.

Problem 5

Let $A\subset X$ be countable and dense in the complete space $X$ without isolated points. Then $A$ is not a $G_\delta$ set.