HW5

From 0506Topology

hide timeline for printing/editing
# Week of... Links (edit)
Fall
1 Sep 12 About, Tue, Thu, Std2Disc
2 Sep 19 Tue, Thu, HW1, 14 Sets
3 Sep 26 Tue, Thu, Photo
4 Oct 3 Tue, Thu
5 Oct 10 HW2, Tue, Thu
6 Oct 17 Tue, Thu, HW3
7 Oct 24 Mon, Tue, Thu
8 Oct 31 Tue, Thu, HW4
9 Nov 7 TE1, Tue, Thu
10 Nov 14 Tue, Thu, HW5
11 Nov 21 Tue, Thu
12 Nov 28 Tue, Thu, HW6
13 Dec 5 Tue, Thu
E Dec 12 TE2
Spring
14 Jan 9 Tue, IT83, Thu, HW7
15 Jan 16 Tue, Thu
16 Jan 23 Tue, HW8, Thu
17 Jan 30 Tue, Thu
18 Feb 6 TE3, Tue, Thu
19 Feb 13 Tue, Thu
R Feb 20
20 Feb 27 Tue, Thu, HW9
21 Mar 6 Tue, Thu, HW10
22 Mar 13 Tue, Thu
23 Mar 20 Tue, Thu, HW11
24 Mar 27 Tue, Thu
25 Apr 3 Tue, Thu, HW12
26 Apr 10 Tue, Thu
Study Apr 17 Office Hours
Exams Apr 24 Final, PM

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
Table of contents
1 Problems for Submission

1.1 45, Problem 1, p. 280
1.2 48, Problem 2, p. 299
1.3 48, Problem 4, p. 299
1.4 48, Problem 6, p. 299

[edit]

Problems for Submission

[edit]

\S45, 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.
[edit]

\S48, 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.
[edit]

\S48, 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.]
[edit]

\S48, Problem 6, p. 299

  1. Show that the irrationals are a Baire space.
Retrieved from "http://katlas.math.toronto.edu/0506-Topology/index.php?title=HW5"