HW3

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Homework Assignment 3

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 5.

Solve the following problems. (But submit only the underlined ones). In Munkres' book, problems 2, 4, 6, 7 (read about "totally disconnected" on page 152), 9 (for X={\mathbb{N}}) and 10 on pages 241-242.

Just for fun.

  1. Compact Hausdorff topologies are "in perfect balance". Indeed, show that if {\mathcal T} is a compact Hausdorff topology on a set X and {\mathcal T}' is either bigger or smaller than {\mathcal T}, then {\mathcal T'}is either not compact or not Hausdorff.
  2. What is the cardinality of \beta{\mathbb N}? Is it countable? As big as the reals? Bigger? Hint. First find surprisingly small dense sets in {\mathbb R}^{\mathbb R} and in [0,1][0,1].

Due date. This assignment is due at the end of our class on Thursday, November 3, 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 38, Problem 2, p. 241
1.2 38, Problem 4, p. 241
1.3 38, Problem 6, p. 242
1.4 38, Problem 7, p. 242
1.5 38, Problem 10, p. 242

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Problems for submission

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\S38, Problem 2, p. 241

  1. Show that the bounded continuous function g \colon (0,1) \to \mathbb R defined by \;g(x) = cos(1/x) cannot be extended to the compactification of Example 3. Define an imbedding h \colon (0,1) \to {[0,1]}^3 such that the functions \;x, \;sin(1/x), and \;cos(1/x) are all extendable to the compactification induced by \;h.
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\S38, Problem 4, p. 241

  1. Let \;Y be an arbitrary compactification of \;X; let \;\beta(X) be the Stone-Cech compactification. Show there is a continuous surjective closed map g \colon \beta(X) \to Y that equals the identity on \;X.
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\S38, Problem 6, p. 242

  1. Let \;X be completely regular. Show that \;X is connected if and only if \;\beta(X) is connected. [Hint: If X = A \cup B is a separation of \;X, let \;f(x) = 0 for x \in A and \;f(x) = 1 for x \in B.]
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\S38, Problem 7, p. 242

Let \;X be a discrete space; consider the space \;\beta(X).

  1. Show that if A \subset X, then \;cl(A) and \;cl(X - A) are disjoint, where the closures are taken in \;\beta(X).
  2. Show that if \;U is open in \;\beta(X), then \;cl(U) is open in \;\beta(X).
  3. Show that \;\beta(X) is totally disconnected.
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\S38, Problem 10, p. 242

We have constructed a correspondence X \to \beta(X) that assigns, to each completely regular space, its Stone-Cech compactification. Now let us assign, to each continuous map f \colon X \to Y of completely regular spaces, the unique continuous map \beta(f) \colon \beta(X) \to \beta(Y) that extends the map i \circ f, where i \colon Y \to \beta(Y) is the inclusion map. Verify the following:

  1. If 1_x \colon X \to X is the identity map of \;X, then \;\beta(1_x) is the identity map of \;\beta(X).
  2. If f \colon X \to Y and g \colon Y \to Z, then \beta(g \circ f) = \beta(g) \circ \beta(f).


These properties tell us that the correspondence we have constructed is what is called a functor; it is a functor from the "category" of completely regular spaces and continuous maps of such spaces, to the "category" of compact Hausdorff spaces and continuous maps of such spaces.

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