  In this article we explore the set of thermal operations from a mathematical
and topological point of view. First we introduce the concept of Hamiltonians
with resonant spectrum with respect to some reference Hamiltonian, followed by
proving that when defining thermal operations it suffices to only consider bath
Hamiltonians which satisfy this resonance property. Next we investigate
continuity of the set of thermal operations in certain parameters, such as
energies of the system and temperature of the bath. We will see that the set of
thermal operations changes discontinuously with respect to the Hausdorff metric
at any Hamiltonian which has so-called degenerate Bohr spectrum, regardless of
the temperature. Finally we find a semigroup representation of the (enhanced)
thermal operations in two dimensions by characterizing any such operation via
three real parameters, thus allowing for a visualization of this set. Using
this, in the qubit case we show commutativity of the (enhanced) thermal
operations as well as convexity of the thermal operations without the closure.
The latter is done by specifying the elements of this set exactly.
