Abstract
Let X, Y be sets with quasiproximities (sic)x and (sic)y (where A (sic) B is interpreted as "B isa neighborhood of A"). Let f.g : X -> Y be a pair of functions such that whenever C (sic)y D. then f(-1) vertical bar C vertical bar (sic)x g(-1)vertical bar D vertical bar. We show that there is then a function h : X -> Y such that whenever C (sic)y D. then f(-1)vertical bar C vertical bar (sic)x h(-1)vertical bar D vertical bar, h(-1)vertical bar C vertical bar (sic)x h(-1)vertical bar D vertical bar and h(-1)vertical bar C vertical bar (sic)x g(-1)vertical bar D vertical bar. Since any function It that satisfies h(-1) vertical bar C vertical bar (sic)x h(-1)vertical bar D vertical bar whenever C (sic)y D, is continuous, many classical "sandwich" or "insertion" theorems are corollaries of this result. The paper is written to emphasize the strong similarities between several concepts
the posets with auxiliary relations studied in domain theory;
quasiproximities and their simplification, Urysohn relations; and
the axioms assumed by Katetov and by Lane to originally show some of these results.
Interpolation results are obtained for continuous posets and Scott domains. We also show that (bi-)topological notions such as normality are captured by these order theoretical ideas. (C) 2010 Elsevier B.V. All rights reserved.
Original language | English |
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Pages (from-to) | 582-593 |
Number of pages | 12 |
Journal | Topology and its Applications |
Volume | 158 |
Issue number | 4 |
DOIs | |
Publication status | Published - 1 Mar 2011 |
Keywords
- Associated order
- Upper semicontinuous
- Bounded complete
- Specialization
- Joincompact
- Uniformly continuous
- Continuous dcpo
- Way below
- PseudoHausdorff
- Monotonically normal
- Approximating
- Bitopological space
- PseudoScott topology
- Order-preserving function
- Lower topology
- Lower semicontinuous
- Scott domain
- Urysohn relation
- (Dualizable) auxiliary relation
- Pairwise continuous function
- Stratifiable
- Scott topology
- Upper and lower adjoint
- Urysohn dual
- Interpolating
- Weakly symmetric
- Extremally disconnected