The exercise is from the handbook of set-theoretic topology page 161: Assume $\mathfrak b=\mathfrak c$. Construct a first countable separable zero-dimensional locally compact pseudocompact space which is not countably compact but which has no uncountable closed discrete subset.

The hint is given: Fisrst construct a space first countable zero-dimensional locally compact space $X$ with underlying set $\mathfrak c$ such that $D=\{\xi: \omega\le \xi < \omega+\omega\}$ is a closed discrete set of cluster points of $\omega$ and such that each countably infinite subset of $X\setminus D$ has a cluster point.

Could anybody help me?

Added: $\mathfrak b$ is the bounding number:

$$\mathfrak b=\min\left\{|F|:F\subseteq{}^\omega\omega\text{ and }\forall f\in{}^\omega\omega\,\exists g\in F\left(g\not\le^* f\right)\right\}\;,$$

where $f\le^*g$ iff there is an $n\in\omega$ such that $f(k)\le g(k)$ for all $k\ge n$. That is, $\mathfrak b$ is the smallest cardinality of an unbounded set in the partial order $\left\langle{}^\omega\omega,\le^*\right\rangle$.


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