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We prove that there exists a countable infinite sequence of non-empty special$\Pi^0_1$ classes $\{\mathcal{P}_i\}_{i\in\omega}$ such that no infinite unionof elements of any $\mathcal{P}_i$ computes the halting set. We then give ageneralized form of lower and upper cone avoidance for infinite unions. Thatis, we show that for any special $\Pi^0_1$ class $\mathcal{P}$ and anycountable sequence of sets in $\mathcal{P}$, $\mathcal{P}$ has a member that isnot computable by the infinite union of elements of the sequence. We also provethe upper cone counterpart, that for any non-recursive set $X$, every non-empty$\Pi^0_1$ class contains a countable sequence of members whose join does notcompute $X$. We finally show that there exists a $\Pi^0_1$ class whose degreespecrum is a countably infinite strict chain.

Countable chains and infinite joins in effectively closed sets of Cantor space