How to show that a Stone Čech compactification of $\mathbb Z_+$ has a cardinality at least $2^\frak c$ Show that $\beta\mathbb Z_+$ (The Stone Čech compactification of the positive integers) has cardinality at least as great as $I^I$ where $I=[0,1]$. 
I know that I is compact Hausdorff and so $I^I$ is compact Hausdorff. I was given a hint that $I^I$ has a countable dense subset but I do not see how that helps. I have also been told that I^I is a compactification of $\mathbb Z_+$ but I don't see how, or how to prove that it is. 
Thanks for the help!!! 
 A: You already have a nice hint. Use it to constuct a surjection from $\beta\mathbb Z_+$ to $I^I$.
A: Any space $X$ has at most $2^{2^{|X|}}$ ultrafilters, then, if $X$ is a Tychonoff space, $|\beta X|\leq 2^{2^{X}}$. For any regular space, one has $\operatorname{w}(X)\leq 2^{\operatorname{d}(X)}$. Thus, for any Tychonoff space $X$, $\operatorname{w}(\beta X)\leq 2^{|X|}$. Now let $\kappa \geq\omega$ be a cardinal and denote the discrete space of cardinality $\kappa$ as $D(\kappa)$. By the  Hewitt-Marczewski-Pondiczery theorem, $I^{2^\kappa}$ has a dense subspace of cardinality $\kappa$; therefore, exists a continuous injection $f: D(\kappa)\to I^{2^{\kappa}}$ such that $f[D(\kappa)]$ is dense in $I^{2^{\kappa}}$. Now, extend $f$ to $\overline{f}:\beta D(\kappa)\to I^{2^\kappa}$. As $\beta D(\kappa)$ is compact, $\overline{f}$ is a continuous surjection from a compact space to a Hausdorff space and therefore $|\beta D(\kappa)|\geq |I^{2^{\kappa}}|$ and $\operatorname{w}(\beta D(\kappa))\geq \operatorname{w}(I^{2^\kappa}) = {2^{\kappa}}$.
As $|\mathbb{Z}_+| = \omega$ and $\operatorname{w}(\mathbb{Z}_+) = \omega$, one has $|\beta \mathbb{Z}_+| = 2^\mathfrak{c}$ and $\operatorname{w}(\beta\mathbb{Z}_+) = \mathfrak{c}$.
A: HINT: 
Let $X$ be any non-empty set, $\tau_d$ the discrete topology on $X$, then for every topology $\tau$, and every function $f\colon X\to X$, $f$ is continuous as a function from $(X,\tau_d)$ to $(X,\tau)$.
Use the fact that $I^I$ has a countable dense set to conclude there is a continuous surjection from $\beta\Bbb Z_+$ onto $I^I$.
