Let $X$ be the real line with the usual topology. Then clearly $|C(X)| = c = |X|$ and on the other hand $|X| = 2^{\aleph\_0}$. 
Now my question is as in the title: **Is there a Tychonoff space $X$ of cardinality not of the form $2^\alpha$ such that $|C(X)| = |X|$** (where $C(X)$ is the set of all real valued continuous functions on $X$). Any reference or help would be appreciated. 



Is there a Tychonoff space $X$ of cardinality not of the form $2^\alpha$ such that $|C(X)| = |X|$
The answer is yes. 


Let $\kappa$ be any singular [strong limit cardinal](http://cantorsattic.info/Strong_limit#strong_limit_cardinal) of uncountable cofinality, such as the cardinal $\beth\_{\omega\_1}$ for a specific example, and let $X=\kappa+1$, the ordinals up to and including $\kappa$ itself. Under the order topology, this is a compact Hausdorff space. Note that $|X|=\kappa\neq 2^\alpha$ for any $\alpha$, since $\kappa$ is a strong limit cardinal. But meanwhile, every continuous function $f:X\to\mathbb{R}$ is eventually constant, in order that it is continuous at the final point $\kappa$, because it has uncountable cofinality. So every function in $C(X)$ is determined by a restriction of it $f\upharpoonright\gamma:\gamma\to\mathbb{R}$ for some $\gamma\lt\kappa$. Since $\kappa$ is a strong limit, there are only $\kappa$ many such restricted functions, and so $|C(X)|=\kappa=|X|$, as desired. 


**Update.** If the GCH fails in a convenient way, we can make a much smaller example. Suppose $2^\omega=\omega\_1$ and $2^{\omega\_1}\gt\omega\_2$, and let $X=\omega\_2+1$. This is a compact Hausdorff space of size $\omega\_2$. Notice that any continuous function $f:X\to\mathbb{R}$ is eventually constant and indeed takes at most countably many distinct values (since otherwise we can find a point $\gamma$ of cofinality $\omega\_1$ with $f\upharpoonright\gamma$ not eventually constant, which will violate continuity at $\gamma$.) It follows that $C(X)$ has size $\omega\_2^\omega$, which has size $\omega\_2$ under our assumptions. So this is a case where $|C(X)|=|X|=\omega\_2$, but $\omega\_2\neq 2^\alpha$ for any $\alpha$. 


