Homework 14: Connectedness

1. Section 6.1, #6.5, 6.8.
2. In this problem we examine Theorem 6.6 from the textbook.
   1. Prove that a topological space \(Z\) is connected if and only if there does not exist a continuous, surjective map \(f:Z\to \{0,1\}\), where \(\{0,1\}\) is given the discrete topology.
   2. Use (a) to give a new proof of Theorem 6.6.
3. Let \(A\) be a subset of a topological space \(X\). Suppose that \(D\subseteq X\) is connected and intersects both \(A\) and \(A^c\). Prove that \(D\) intersects \(\partial A\).
   Hint: Recall that \(X=A^{\circ}\ \dot{\cup}\ \partial A\ \dot{\cup}\ (A^c)^{\circ}\) (the disjoint union of the three sets).
4. (Real Analysis) Let \(X\) denote the subset of \(\bR^{\infty}\) consisting of sequences \(\bx=(x_1,x_2,x_3,\ldots)\) such that \(\sum x_i^2\) converges. You may use without proof any standard facts about infinite series.
   1. Prove that if \(\bx, \by\in X\), then \(\sum |x_iy_i|\) converges. (Hint: Use Problem 4(b) from Homework 13 to show that the partial sums are bounded.)
   2. Let \(c\in \bR\). Prove that if \(\bx, \by \in X\), then \(\bx+\by, c\bx\in X\).
   3. Prove that $$ d(\bx,\by)=\left[\sum_{i=1}^{\infty}(x_i-y_i)^2\right]^{1/2} $$ is a well-defined metric on \(X\). It is called the \(\ell^2\) metric.