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.

翻译：我们以建设性的方式工作,我们研究连续的完整表层(dcpos)和斯科特表层学。 我们的两个主要新颖之处是内在分化的概念和尖锐元素的概念。 分开是不平等的正面表述, 类似于住人的方式是非空的正面表述。 举例来说, 一个更低的底部是锐利的, 并且只有当它被定位的时候, 我们注意到一个更低的底部是尖锐的。 我们的第一个主要结果就是对于一大批连续的 dcpos, 桥梁- V ⁇ i\c{t ⁇ v{a} 差异表层学和斯科特表层学是同时的。 虽然我们无法期望在非三角的表层上出现紧密或共同的分化。 我们虽然不能期望在非三角的表层上出现紧密或一致的分化。 我们证明, 内在的分化是紧密的分化和相互交叉的, 当它被限制在连续的 dcpo 的尖锐元素上时, 包括强烈的最大化元素, 正如Smy 和 Hecmann 所研究的那样。 我们开发了强烈的理论, 最优化的理论, 强调它与尖锐和劳森的表层的分层。 最后, 我们通过考虑一个不断的自然的平整的地、 的平整的空空间的地、 直成一个自然的地的地的地、 直成一个自然的地的地的地的地的分层。