###### topological space

- In the branch of mathematics known as topology, the specialization (or canonical) preorder is a natural preorder on the set of the points of a topological space. (wikipedia.org)
- Consider any topological space X. The specialization preorder ≤ on X relates two points of X when one lies in the closure of the other. (wikipedia.org)
- Let X be a topological space and let ≤ be the specialization preorder on X. Every open set is an upper set with respect to ≤ and every closed set is a lower set. (wikipedia.org)
- For any topological space satisfying the T0 separation axiom, the topology induces an order relation on that space, the specialization order: x ≤ y if and only if every open neighbourhood of x is also an open neighbourhood of y. (wikipedia.org)

###### topology

- A directed complete partial order (dcpo) with the Scott topology is always a Kolmogorov space (i.e., it satisfies the T0 separation axiom). (wikipedia.org)
- The order relation of a dcpo D can be reconstructed from the Scott-open sets as the specialization order induced by the Scott topology. (wikipedia.org)
- However, a dcpo equipped with the Scott topology need not be 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. (wikipedia.org)

###### axiom

- For most spaces that are considered in practice, namely for all those that satisfy the T0 separation axiom, this preorder is even a partial order (called the specialization order). (wikipedia.org)
- Therefore, the antisymmetry of ≤ is precisely the T0 separation axiom: if x and y are indistinguishable then x = y. (wikipedia.org)
- On the other hand, the symmetry of specialization preorder is equivalent to the R0 separation axiom: x ≤ y if and only if x and y are topologically indistinguishable. (wikipedia.org)

###### clear separation

- The spec does have a clear separation of the various aspects of the protocol. (w3.org)
- There is no clear separation between sensory-motor and cognitive specializations underpinning such skills, undermining the notion of executive control as a distinct process. (dur.ac.uk)

###### partial

- The Singapore system of government, as with those of a number of other Commonwealth jurisdictions, exhibits a partial separation of powers. (wikipedia.org)
- The key notion that exemplifies "partial" cloning from "classical" cloning is the separation of the mechanism(s) that "wipe clean" the specialization of a cell from those that "wipe-clean" the age of the cell. (wikipedia.org)

###### order

- The specialization order is often considered in applications in computer science, where T0 spaces occur in denotational semantics. (wikipedia.org)
- The specialization order is also important for identifying suitable topologies on partially ordered sets, as it is done in order theory. (wikipedia.org)
- In the Sierpinski space {0,1} with open sets {∅, {1}, {0,1}} the specialization order is the natural one (0 ≤ 0, 0 ≤ 1, and 1 ≤ 1). (wikipedia.org)
- In this case it is justified to speak of the specialization order. (wikipedia.org)
- IMS instruments are extremely sensitive stand-alone devices, but are often coupled with mass spectrometry, gas chromatography or high-performance liquid chromatography in order to achieve a multi-dimensional separation. (wikipedia.org)
- Perhaps ion mobility spectrometry's greatest strength is the speed at which separations occur-typically on the order of tens of milliseconds. (wikipedia.org)

###### idea

- Specialization as an idea is applied also in valuation theory. (wikipedia.org)
- Judicial independence allows the judiciary to check the exercise of power by the other branches of government more effectively, enhancing the idea of the separation of powers. (wikipedia.org)
- The idea of separation of powers is founded on the basis of constitutionalism. (wikipedia.org)

###### cell

- These compartments allow the separation/specialization of processes within the cell. (edu.au)

###### design

- Visualization Library design is based on algorithmic and data structure specialization and separation, unlike many other 3D frameworks part of the so-called "uber scene graph" family, that is, those 3d engines that keep all the rendering information in a single hierarchical structure. (wikipedia.org)

###### open

- These restatements help to explain why one speaks of a "specialization": y is more general than x, since it is contained in more open sets. (wikipedia.org)

###### Health

- Spillover Effects of Job Separations: Does Becoming Unemployed Among Youth Affect Health of Their Family Members? (diva-portal.org)