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could anyone help to show that $[0,1]^{\mathbb{N}}$ with respect to the box topology is not compact? Thank you!

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3 Answers 3

up vote 8 down vote accepted

There’s absolutely nothing wrong with showing non-compactness of $X = \square_{k=0}^\infty [0,1]$ directly by looking at open covers, but there are other ways as well. For instance:

If $X = {\Large \square}_{k=0}^\infty [0,1]$ were compact, its closed subspace $ {\Large \square}_{k=0}^\infty \{0,1\}$ would be compact, but it’s not hard to show that $ {\Large \square}_{k=0}^\infty \{0,1\}$ is an infinite, closed, discrete set in $X$ and therefore cannot be compact.

Even simpler:

For $n\in\mathbb{N}$ let $x_n \in X$ be the point such that $x_n(n) = 1$ and $x_n(k) = 0$ if $k\ne n$. Now consider the set $A = \{x_n:n\in\mathbb{N}\}$. It’s infinite, so if $X$ were compact, $A$ would have a limit point in $X$. But it’s not hard to show that $A$ is a closed, discrete subset of $X$ and therefore has no limit point in $X$.

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Just so nobody else gets confused like I did, the square box means "Cartesian product with the box topology", not some kind of missing or mis-rendered character in your browser. –  Ted Aug 28 '11 at 6:42
    
@Ted: Sorry about that; I used to work a bit with box products and got used to that notation. –  Brian M. Scott Aug 28 '11 at 8:36

Let $a=(a_1,a_2,\dots)$ be a sequence consisting of only $0$'s and $1$'s and define $U_{an}=[0,\frac{3}{4})$ if $a_n=0$ and $U_{an}=(\frac{1}{4},1]$ if $a_n=1$. Also, define $U_a=\prod_{n=1}^{\infty} U_{an}$.

Exercise 1: Prove that $U_a$ is an open subset of $[0,1]^{\mathbb{N}}$ in the box topology but that it is not an open subset of $[0,1]^{\mathbb{N}}$ in the product topology.

Exercise 2: Prove that the collection of subsets of $[0,1]^{\mathbb{N}}$ of the form $U_a$ where $a=(a_1,a_2,\dots)$ is a sequence consisting of only $0$'s and $1$'s is an open cover of $[0,1]^{\mathbb{N}}$ in the box topology.

Exercise 3: Does this open cover have a finite subcover?

The following problems should be of additional interest:

Problem 1: Let $\{X_n\}_{n\in\mathbb{N}}$ be a collection of topological spaces. If the box topology on $\prod_{n=1}^{\infty} X_n$ is a compact topological space, what can you deduce about the $X_n$'s? In other words, how do the topologies on the individual $X_n$'s affect the compactness of $\prod_{n=1}^{\infty} X_n$ in the box topology?

Problem 2: In the context of Problem 1, if $\prod_{n=1}^{\infty} X_n$ is compact in the box topology, then prove that $X_n$ is compact for all $n\in\mathbb{N}$.

Problem 3: A topological space $X$ is said to be countably compact if every open cover of $X$ has a countable subcover. Is $[0,1]^{\mathbb{N}}$ countably compact in the box topology?

I hope this helps!

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Problem 1 is best set in the context of $T_1$-spaces; otherwise a complete answer gets a little messy. Assuming that one already has the Tikhonov theorem, Problem 2 has an easy solution that doesn’t require knowing the result of Problem 1. (Not a complaint, just an observation.) –  Brian M. Scott Aug 28 '11 at 8:57
    
@Brian Of course, you are right. I did not suggest that Problem 2 requires Problem 1; I only suggested that the notation of Problem 2 is borrowed from Problem 1 (thus, "In the context of Problem 1"). Also, I do not think one needs Tikhonov's theorem to solve Problem 2; one can note that the projection map $\prod_{n=1}^{\infty} X_n\to X_i$ is continuous for all $i\in\mathbb{N}$. Problem 1 is more of an open-ended problem; the OP is free to interpret the problem whichever way he/she chooses. If he/she chooses to assume the $T_1$-axiom, then this is fine. –  Amitesh Datta Aug 28 '11 at 23:09

$$\left\{\displaystyle\prod_{n\in \mathbb{N}} I_n : I\in \left\{[0,\frac23),(\frac13,1]\right\}^\mathbb{N}\right\}$$

is an open cover of $[0,1]^{\mathbb{N}}$ with the box topology that does not have a finite subcover.

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