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Reverse Mathematics (RM hereafter) is a program in the foundations ofmathematics founded by Friedman and developed extensively by Simpson andothers. The aim of RM is to find the minimal axioms needed to prove a theoremof ordinary, i.e. non-set-theoretic, mathematics. As suggested by the title,this paper deals with the study of the topological notions of dimension andparacompactness, inside Kohlenbach's higher-order RM. As to splittings, thereare some examples in RM of theorems $A, B, C$ such that$A\leftrightarrow(B\wedge C)$, i.e. $A$ can be split into two independent(fairly natural) parts $B$ and $C$, and the aforementioned topological notionsgive rise to a number of splittings involving highly natural $A, B, C$.Nonetheless, the higher-order picture is markedly different from thesecond-one: in terms of comprehension axioms, the proof in higher-order RM ofe.g. the paracompactness of the unit interval requires full second-orderarithmetic, while the second-order/countable version of paracompactness of theunit interval is provable in the base theory of second-order RM. We obtainsimilarly 'exceptional' results for the Urysohn identity, the Lindel\"of lemma,and partitions of unity. We show that our results exhibit a certain robustness,in that they do not depend on the exact definition of cover, even in theabsence of the axiom of choice.

Reverse Mathematics of topology: dimension, paracompactness, and splittings