What are some applications of large cardinals? Most mathematicians don't often encounter cardinalities larger than that of the continuum, it seems? What are some results outside of pure set theory/logic that rely on the properties of larger cardinals? In a similar vein, can anyone give examples of types of mathematical objects (groups, fields, etc.) that have cardinalities larger than the continuum?

Some objects whose cardinal is at least $2^c,$ where $c$ is the cardinal of the reals : (1). The set of all real functions. (2). The set of all filters on an infinite set. (3). The dual space $l_{\infty}^*$ of the Banach space $l_{\infty}.$ (4). The maximal compactifications of $N$,of $Q$,and of $R$. (5).  The free group on any set $S$ such that $|S|>c.$
One result that comes to mind is that if $X$ is a separable Tychonoff space and $X$ has a closed discrete subspace $Y$ with $|Y|\geq c$ then $X$ is not a normal space. This relies on the facts that the power set of $Y$ has cardinal at least $2^c$ and that the set of continuous $g:X\to R$ has cardinal at most $c$, and that $c<2^c$. Examples: (1). The Niemitzky plane is not normal. (2).  $bN$  has a non-normal subspace, where $bN$ is the maximal compactification of $N.$
