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## Due October 19

### 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. If X doesn't have isolated points, 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 $\bigcup_n \text{int}(F_n)$ is dense in X.

### Problem 3

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.

### Problem 4

Let $d(x) = d(x,\Z)$ denote the distance from $x\in\R$ to the nearest integer. For $q\in\Z_+, \alpha>0$, define the sets

$U_\alpha(q) = \{x\in\R: d(qx)< q^{-\alpha}\}$

and

$Y_\alpha = \{x\in\R: x$ belongs to infinitely many $U_\alpha(q)\}$.

1. $Y_\alpha$ is a $G_\delta$ subset of $\R$

2. $X = \bigcap_{\alpha>0} Y_\alpha$ is a dense $G_\delta$ subset of $\R$.

3. For each $x\in\R$, $x\not\in X$ iff there exists a polynomial p over $\R$ such that $p(n)d(nx)>1$ for all $n\ge1$.
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