HW2

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Homework Assignment 2

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 chapter 3.

Solve the following problems. (But submit only the underlined ones). In Munkres' book (Topology, 2nd edition), problems 6, 7 on page 152, problems 1, 2, 3, 8, 10 on pages 157-158 and problems 1, 4, 5, 7, 8 on pages 170-171.

Just for fun.

  1. Can you find a connected subset W of {\mathbb R}^2 which is pathwise totally disconnected (i.e., there is no x\neq y\in W that can be connected via a continuous path)?
  2. Without referring to the famed "Tychonoff Theorem", prove that [0,1]^{\mathbb N} is compact in the product topology. (Is it compact in the box topology?)

Due date. This assignment is due at the end of our class on Thursday, October 20, 2005. If we will remain behind schedule Dror may postpone the due date or cancel some of the problems; please stay tuned.

Putnam competition. 

Date: Sun, 25 Sep 2005 12:58:55 -0400

To: Lecturers in mathematics and engineering science

From: Ed Barbeau

Re: Putnam competition

The Putnam Competition will be written on the first Saturday
in December (Dec. 3, 2005). It consists of two three hour
examinations with six problems apiece.

Could you please make this known to your specialist, major and
engineering science classes, along with any other students that
might be interested and qualified. In particular, I would
appreciate your encouraging first year students to participate.
(I do not have any direct contact with them yet.)

Registration is free. Students can simply send me an email
message, telling me their year and program: barbeau@math.utoronto.ca .

Many thanks. Ed Barbeau
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 23, Problem 6, p. 152
1.2 24, Problem 1, p. 158
1.3 24, Problem 10, p. 158
1.4 26, Problem 1, p. 170
1.5 26, Problem 4, p. 171
1.6 26, Problem 5, p. 171

[edit]

Problems for submission

[edit]

\S23, Problem 6, p. 152

  1. Let A \subset X. Show that if C is a connected subspace of X that intersects both A and X − A, then C intersects bdA.
[edit]

\S24, Problem 1, p. 158

  1. Show that no two of the spaces (0,1), (0,1], and [0,1] are homeomorphic. (Hint: What happens if you remove a point from each of these spaces?)
  2. Suppose that there exist imbeddings f \colon X \to Y and g \colon Y \to X. Show by means of an exmaple that X and Y need not be homeomorphic.
  3. Show {\mathbb R}^n and \mathbb R are not homeomorphic if n > 1.
[edit]

\S24, Problem 10, p. 158

  1. Show that if U is an open connected subspace of {\mathbb R}^2, then U is path connected. [Hint: Show that given x_0 \in U, the set of points that can be joined to x0 by a path in U is both open and closed in U.
[edit]

\S26, Problem 1, p. 170

  1. Let \mathcal{T} and \mathcal{T'} be two topologies on the set X; suppose that \mathcal{T'} \supset \mathcal{T}. What does compactness of X under one of these topologies imply about compactness under the other?
  2. Show that if X is compact Hausdorff under both \mathcal{T} and \mathcal{T'}, then either \mathcal{T} and \mathcal{T'} are equal or they are not comparable.
[edit]

\S26, Problem 4, p. 171

  1. Show that every compact subspace of a metric space is bounded in that metric and is closed. Find a metric space in which not every closed bounded subspace is compact.
[edit]

\S26, Problem 5, p. 171

  1. Let A and B be disjoint compact subspaces of the Hausdorff space X. Show that there exist disjoint open sets U and V containing A and B, respectively.
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