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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](http://math.stackexchange.com/questions/97413/on-the-product-of-separable-spaces).2012-04-21
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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.2012-04-21
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    Perhaps you should specify what is your definition of [Stone-Cech compactification](http://en.wikipedia.org/wiki/Stone%E2%80%93%C4%8Cech_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.2012-04-21
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    @martini Do you mean that $\left|[0,1]^{\frak c}\right|=(2^{\aleph_0})^{\frak c} = 2^{\aleph_0\cdot\mathfrak c}=2^\frak c$?2012-04-21
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    For the thing about cardinality of $[0,1]^{\mathfrak c}$ see also [here](http://math.stackexchange.com/questions/57389/how-to-prove-cardinality-equality-mathfrak-c-mathfrak-c-2-mathfrak-c).2012-04-21
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    @AsafKaragila You're right.2012-04-22
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    For the first part you can also have a look at this question: [About Cardinality in Hausdorff Spaces](http://math.stackexchange.com/questions/45429/about-cardinality-in-hausdorff-spaces).2012-05-05

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