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This is my extra credit homework problem and I have no idea how to prove this please help me. Thank you.

We denote by #S the cardinality of a set S. $\aleph_0$ = #$\mathbb{N}$, $c=2^{\aleph_0}$ = #$\mathbb{R}$. Let X be a topological space and let Y be a dense subset of X. Prove that #X $\le$ $2^{2^{\#Y}}$. Conclude that $\#X \le 2^c$ whenever X is separable.

Also, how to prove the product space $[0,1]^c$ (equipped with the product topology) is separable and $\# \beta \mathbb{N}$ = $2^c$.

First, I construct a continuous surjection $\# \beta \mathbb{N}$ $\rightarrow$ $[0,1]^c$. then what is the cardinality of $[0,1]^c$?

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For separability of $[0,1]^{\mathfrak c}$ you could use result from this question: On the product of separable spaces. –  Martin Sleziak Apr 21 '12 at 5:01
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The first statement is false as stated, you need some assumption on your topological space. For example, let $X$ be any (nonempty) set with the indiscrete topology and let $Y$ be any point of $X$. Then $\overline{Y} = X$ since it's the only closed set containing $Y$, so $Y$ is dense. –  Jason DeVito Apr 21 '12 at 5:03
    
Perhaps you should specify what is your definition of Stone-Cech compactification. AFAIK two definitions are used quite frequently. $\beta\mathbb N$ can be defined by the property, that it contains $\mathbb N$ as a subspace and every map $\mathbb N\to[0,1]$ can be uniquely extended to a continuous map on $\beta\mathbb N$. Quite frequently description of $\beta\mathbb N$ as the of all ultrafilters on $\mathbb N$ with the topology generated by the sets $\widehat A=\{\mathcal F; A\in\mathcal F\}$ is used. –  Martin Sleziak Apr 21 '12 at 5:16
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@martini Do you mean that $\left|[0,1]^{\frak c}\right|=(2^{\aleph_0})^{\frak c} = 2^{\aleph_0\cdot\mathfrak c}=2^\frak c$? –  Asaf Karagila Apr 21 '12 at 12:31
    
For the thing about cardinality of $[0,1]^{\mathfrak c}$ see also here. –  Martin Sleziak Apr 21 '12 at 13:55

2 Answers 2

I am never sure how much should I write, when the question is homework. (I want to give hints but not to give away too much.) But since this is obviously available in many textbooks where the OP could find it easily, I think it does not matter that much how detailed hints I'll give.


  • Hint for the estimate of a cardinality of space: Do you know result saying that if $x\in\overline Y$ then there is a filter base consisting of subsets of $Y$ that converges to $x$? What is maximal possible cardinality of such filter base? In combination with the assumption that $X$ is Hausdorff (which seems to be missing), can you get an estimate for the cardinality of $X$?

  • For separability of $[0,1]^{\mathfrak c}$ you could use result from this question: On the product of separable spaces.

  • Since $\beta\mathbb N$ is compact, any continuous dense map must be a surjection. (Why?) Can you use the fact that $[0,1]^{\mathfrak c}$ is separable and the definition of the Stone-Čech compactification $\beta\mathbb N$ to get such a map?

  • By existence of surjective map $|\beta\mathbb N|\ge |[0,1]^{\mathfrak c}|=2^{\mathfrak c}$. Since $\beta\mathbb N$ contains $\mathbb N$ as a dense subset, you get $|\beta\mathbb N|\le 2^{\mathfrak c}$ from the first part.


You can find this result in several texts on general topology, e.g. the following:

  • Theorem 3.6.11 in Engelking: General Topology, Heldermann 1989. (A more general statement about cardinality of the Stone-Čech compactification of arbitrary discrete space is given here.0

  • Example 19.13 d) in Willard: General Topology, p.140

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The first question requires that $X$ be Hausdorff. If $X$ is any set, no matter how large, it can be equipped with the cofinite topology, which makes it a separable $T_1$-space. (Indeed, every countably infinite subset will be dense.) You don't actually need to know anything about filter bases and their convergence. Let $D$ be a dense subset of $X$. For each $x\in X$ let $\mathscr{N}(x)$ be the family of all open sets containing $x$, and let $\mathscr{D}(x)=\{V\cap D:V\in\mathscr{N}(x)\}$. Use the fact that $X$ is Hausdorff to show that $\mathscr{D}(x)\ne\mathscr{D}(y)$ whenever $x,y\in X$ with $x\ne y$. Then note that $\mathscr{D}(x)\subseteq\wp(D)$ for each $x\in X$. What is $|\wp(D)|$? How many subsets does $\wp(D)$ have?

I've nothing to add to Martin's comments on the other questions.

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