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Homework 7, Real Analysis

Due Date: September 22

Problem 1

Let X be a complete metric space.

  1. The countable intersection of dense $latex G_\delta$ sets in X is a dense $latex G_\delta$ set in X.

  2. If a set and its complement are dense subsets of X, at most one can be $latex G_\delta$.

  3. A countable dense subset of X cannot be $latex G_\delta$.

Problem 2

Let X be a complete metric space.

  1. If $latex O\subset X$ is open, then O is a metric subspace of the second category.

  2. If $latex \{F_n\}$ are closed subsets of X with $latex X = \bigcup_n F_n$, then $latex \bigcup_n F_n$ is dense in X.

Problem 3

Let $latex \{f_n\}\subset C(\R)$ be such that for each $latex x\in\R$ there exists $latex n\ge 1$ such that $latex f_n(x)=0.$ Let O be the set of $latex x\in\R$ such that there exist $latex n\ge 1$ and $latex \e>0$ such that $latex f_n|_{(x-\e,x+\e)}=0$. Then O is an open dense set in $latex \R$.

Problem 4

Let $latex f:\R\to\R$ be an infinitely differentiable function such that for all $latex x\in\R$ there exists n such that $latex f^{(n)}(x) = 0$. Then f is a polynomial.

Problem 5

Let X be a complete metric space.

  1. The set of points of discontinuity of the characteristic function $latex \chi_O$ of an open set O is a nowhere dense subset of X.

  2. Given open sets $latex \{O_n\}$, there exists $latex x\in X$ such that $latex \chi_{O_n}$ is continuous at x for each n.