# HW5

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Homework Assignment 5

Required reading. Read, reread and rereread your notes to this point, and make sure that you really, really really, really really really understand everything in them. Do the same every week! Also, read all of Munkres sections 45, 48 and 49.

Solve the following problems. (But submit only the underlined ones). In Munkres' book, page 280 problems 1 and 2 and page 299 problems 2, 3, 4, 6 and 8.

Due date. This assignment is due at the end of our class on Thursday, December 1, 2005.

 Above the line, edit only if you are sure Feel free to edit below the line

## Problems for Submission

### $\S$45, Problem 1, p. 280

1. If Xn is metrizable with metric dn, then $D(\mathbf{x}, \mathbf{y}) = \sup\left\{\bar{d_i}(x_i, y_i)/i \right\}$ is a metric for the product space $X = \prod X_n$. Show that X is totally bounded under D if each Xn is totally bounded under dn. Conclude without using the Tychonoff theorem that a countable product of compact metrizable spaces is compact.

### $\S$48, Problem 2, p. 299

1. The Baire category theorem implies that $\mathbb{R}$ cannot be written as a countable union of closed subsets having empty interiors. Show this fails if the sets are not required to be closed.

### $\S$48, Problem 4, p. 299

1. Show that if every point x of X has a neighborhood that is a Baire space, then X is a Baire space. [Hint: Use the open set formulation of the Baire condition.]

### $\S$48, Problem 6, p. 299

1. Show that the irrationals are a Baire space.