Homework 9, Real Analysis

Due October 19

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

Problem 3

Let X be a complete metric space.

1. The set of points of discontinuity of the characteristic function $latex \chi_O$ 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.

Problem 4

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

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

and

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

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


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


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