Let $ X$ be a topological space. The set $C(X)$ is all continuous, real valued functions on $X$, so that $C(X) \subseteq \mathbb{R}^{X}$.
Define:
$C^* (X) = \{ f \in C(X) \vert \quad f\quad is \quad bounded \}$.
consider the subspace $\mathbb{R}$ , $ \mathbb{Q}$, $\mathbb{N}$ and $ \mathbb{N}^{*} = \{ 1, 1/2, .........., 1/n ,...... 0 \} $ of $ \mathbb{R}$ , and the ring $ C $ and $ C ^{*}$ for each of these spaces. each of these rings is of cardinal $ c$.
Will you help me solve these problems?
a:$ C ( \mathbb{R} )$ has just two idempotents, $ \mathbb{N}^{*}$ has exactly $\aleph_{0}$, and $ C ( \mathbb{Q} )$ and $ C ( \mathbb{N} )$ have cardinal $ c$.
b: every nonzero idempotent in $ C ( \mathbb{Q} )$ is a sum of two nonzero idempotents. in $ C ( \mathbb{N} )$, and in $ C ( \mathbb{N^{*}} )$,some, but not all idempotents have this property.
c: Except for the obvious identity $C ( \mathbb{N}^{*} ) = C^{*} ( \mathbb{N}^{*} )$ , no two of the rings in question are isomorphic.