Homework 9, Real Analysis

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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