Ideal chain in $\mathbb{R},\mathbb{Q}, \mathbb{N}$

2017-12-13 13:57:07

An ideal $I$ in $C(X)$ is called $z$-ideal if $Z(f) \in Z(I) $ implies $f \in I$.

$Z(f)=\{ x \in X : f(x) = 0 \} $

Help me solve the following problem?

1: Find a chain of $z$ -ideals in $C(\mathbb{R})$ ( under set

inclusion ) that is in one - one ,order-pereserving corresponding with

$\mathbb{R}$ itself.

2: Find a chain of $z$ -ideals in $C(\mathbb{Q})$ is one - one

,order-pereserving corresponding with $\mathbb{R}$.

Do the same for $C(\mathbb{N})$?