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Under $\text{CH}$ we construct a partition of Baire space into compact sets,which is indestructible by countably supported iteration and product of Sacksforcing of any length, answering a question of Newelski. Further, we present anin-depth isomorphism-of-names argument for $\text{spec}(\mathfrak{a}_\text{T})= \{\aleph_1, \mathfrak{c}\}$ in the product-Sacks model. Finally, we provethat Shelah's ultrapower model for the consistency of $\mathfrak{d} <\mathfrak{a}$ also satisfies $\mathfrak{a} = \mathfrak{a}_\text{T}$. Thus,consistently $\aleph_1 < \mathfrak{d} < \mathfrak{a} = \mathfrak{a}_\text{T}$may hold.

Partitions of Baire space into compact sets