  Working constructively, we study continuous directed complete posets (dcpos)
and the Scott topology. Our two primary novelties are a notion of intrinsic
apartness and a notion of sharp elements. Being apart is a positive formulation
of being unequal, similar to how inhabitedness is a positive formulation of
nonemptiness. To exemplify sharpness, we note that a lower real is sharp if and
only if it is located. Our first main result is that for a large class of
continuous dcpos, the Bridges-V\^i\c{t}\v{a} apartness topology and the Scott
topology coincide. Although we cannot expect a tight or cotransitive apartness
on nontrivial dcpos, we prove that the intrinsic apartness is both tight and
cotransitive when restricted to the sharp elements of a continuous dcpo. These
include the strongly maximal elements, as studied by Smyth and Heckmann. We
develop the theory of strongly maximal elements highlighting its connection to
sharpness and the Lawson topology. Finally, we illustrate the intrinsic
apartness, sharpness and strong maximality by considering several natural
examples of continuous dcpos: the Cantor and Baire domains, the partial
Dedekind reals, the lower reals and finally, an embedding of Cantor space into
an exponential of lifted sets.
