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## Due Date: September 22

### Problem 1

Let X be a complete metric space.

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

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

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

### Problem 2

Let X be a complete metric space.

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

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

### Problem 3

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

### Problem 4

Let $f:\R\to\R$ be an infinitely differentiable function such that for all $x\in\R$ there exists n such that $f^{(n)}(x) = 0$. 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 $\chi_O$ of an open set O is a nowhere dense subset of X.

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