  To enable the study of open sets in computational approaches to mathematics,
lots of extra data and structure on these sets is assumed. For both
foundational and mathematical reasons, it is then a natural question, and the
subject of this paper, what the influence of this extra data and structure is
on the logical and computational properties of basic theorems pertaining to
open sets. To answer this question, we study various basic theorems of
analysis, like the Baire category, Heine, Heine-Borel, Urysohn, and Tietze
theorems, all for open sets given by their (third-order) characteristic
functions. Regarding computability theory, the objects claimed to exist by the
aforementioned theorems undergo a shift from `computable' to `not computable in
any type two functional', following Kleene's S1-S9. Regarding Reverse
Mathematics, the latter's so-called Main Question, namely which set existence
axioms are necessary for proving a given theorem, does not have a unique or
unambiguous answer for the aforementioned theorems, working in Kohlenbach's
higher-order framework. A finer study of representations of open sets leads to
the new `$\Delta$-functional' which has unique (computational) properties.
