Editing Scott continuity (section)

==Properties==
A Scott-continuous function is always [[monotone function|monotonic]], meaning that if <math>A \le_{P} B</math> for <math>A, B \subset P</math>, then <math>f(A) \le_{Q} f(B)</math>.

A subset of a directed complete partial order is [[closed set|closed]] with respect to the Scott topology induced by the partial order if and only if it is a [[lower set]]  and closed under suprema of directed subsets.<ref name="AbramskyJung1994"/>

A [[directed complete partial order]] (dcpo) with the Scott topology is always a [[Kolmogorov space]] (i.e., it satisfies the [[T0 separation axiom|T<sub>0</sub> separation axiom]]).<ref name="AbramskyJung1994"/> However, a dcpo with the Scott topology is a [[Hausdorff space]] if and only if the order is trivial.<ref name="AbramskyJung1994"/> The Scott-open sets form a [[complete lattice]] when ordered by [[inclusion (set theory)|inclusion]].<ref name="BauerTaylor2009"/>

For any Kolmogorov space, the topology induces an order relation on that space, the [[specialization order]]: {{nowrap|''x'' ≤ ''y''}} if and only if every [[open neighbourhood]] of ''x'' is also an open neighbourhood of ''y''. The order relation of a dcpo ''D'' can be reconstructed from the Scott-open sets as the specialization order induced by the Scott topology. However, a dcpo equipped with the Scott topology need not be [[sober space|sober]]: the specialization order induced by the topology of a sober space makes that space into a dcpo, but the Scott topology derived from this order is finer than the original topology.<ref name="AbramskyJung1994">{{cite book |last1=Abramsky |first1=S. |last2=Jung |first2=A. |editor1-first=S. |editor1-last=Abramsky |editor2-first=D.M. |editor2-last=Gabbay |editor3-first=T.S.E. |editor3-last=Maibaum |title=Handbook of Logic in Computer Science |volume=III |year=1994 |publisher=Oxford University Press |isbn=978-0-19-853762-5 |chapter=Domain theory |chapter-url=http://www.cs.bham.ac.uk/~axj/pub/papers/handy1.pdf }}</ref>