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Tarea 8, Análisis real

Due October 2nd

Problem 1

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

Problem 2

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

Problem 3

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

Problem 4

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

Problem 5

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