cardinality of a set X in a topological space 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 $[0,1]^c$? 

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.
