  Reverse Mathematics (RM hereafter) is a program in the foundations of
mathematics founded by Friedman and developed extensively by Simpson and
others. The aim of RM is to find the minimal axioms needed to prove a theorem
of ordinary, i.e. non-set-theoretic, mathematics. As suggested by the title,
this paper deals with the study of the topological notions of dimension and
paracompactness, inside Kohlenbach's higher-order RM. As to splittings, there
are 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 notions
give rise to a number of splittings involving highly natural $A, B, C$.
Nonetheless, the higher-order picture is markedly different from the
second-one: in terms of comprehension axioms, the proof in higher-order RM of
e.g. the paracompactness of the unit interval requires full second-order
arithmetic, while the second-order/countable version of paracompactness of the
unit interval is provable in the base theory of second-order RM. We obtain
similarly '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 the
absence of the axiom of choice.
