## Interposition and lattice cones of functions

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- by Jörg Blatter and G. L. Seever PDF
- Trans. Amer. Math. Soc.
**222**(1976), 65-96 Request permission

## Abstract:

A lattice cone of functions on a set*X*is a convex cone of bounded real-valued functions on

*X*which contains the constants and which is closed under the lattice operations. Our principal results concern the relation between closed lattice cones on a set

*X*and certain binary relations, called inclusions, on the power set of

*X*. These results are applied to interposition problems, Császár compactifications of quasi-proximity spaces, the compactification of Nachbin’s completely regular ordered topological spaces, and a problem in best approximation.

## References

- Jörg Blatter,
*Grothendieck spaces in approximation theory*, Memoirs of the American Mathematical Society, No. 120, American Mathematical Society, Providence, R.I., 1972. MR**0493107** - J. Blatter and G. L. Seever,
*Interposition of semi-continuous functions by continuous functions*, Analyse fonctionnelle et applications (Comptes Rendus Colloq. d’Analyse, Inst. Mat., Univ. Federal Rio de Janeiro, Rio de Janeiro, 1972), Actualités Sci. Indust., No. 1367, Hermann, Paris, 1975, pp. 27–51. MR**0435793** - J. Blatter and G. L. Seever,
*Inclusions, interposition and approximation*, Approximation theory (Proc. Internat. Sympos., Univ. Texas, Austin, Tex., 1973) Academic Press, New York, 1973, pp. 257–261. MR**0333531**
—, - Ákos Császár,
*Foundations of general topology*, A Pergamon Press Book, The Macmillan Company, New York, 1963. MR**0157340** - C. H. Dowker,
*On countably paracompact spaces*, Canad. J. Math.**3**(1951), 219–224. MR**43446**, DOI 10.4153/cjm-1951-026-2 - V. A. Efremovič,
*Infinitesimal spaces*, Doklady Akad. Nauk SSSR (N.S.)**76**(1951), 341–343 (Russian). MR**0040748** - V. A. Efremovič,
*The geometry of proximity. I*, Mat. Sbornik N. S.**31(73)**(1952), 189–200 (Russian). MR**0055659** - M. Katětov,
*On real-valued functions in topological spaces*, Fund. Math.**38**(1951), 85–91. MR**50264**, DOI 10.4064/fm-38-1-85-91 - M. Katětov,
*Correction to “On real-valued functions in topological spaces” (Fund. Math. 38 (1951), pp. 85–91)*, Fund. Math.**40**(1953), 203–205. MR**60211**, DOI 10.4064/fm-40-1-203-205 - B. R. Kripke and R. B. Holmes,
*Approximation of bounded functions by continuous functions*, Bull. Amer. Math. Soc.**71**(1965), 896–897. MR**182702**, DOI 10.1090/S0002-9904-1965-11433-1 - Bernard Kripke and Richard Holmes,
*Interposition and approximation*, Pacific J. Math.**24**(1968), 103–110. MR**221178**, DOI 10.2140/pjm.1968.24.103 - Ernest Michael,
*Continuous selections. I*, Ann. of Math. (2)**63**(1956), 361–382. MR**77107**, DOI 10.2307/1969615 - Leopoldo Nachbin,
*Topology and order*, Van Nostrand Mathematical Studies, No. 4, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1965. Translated from the Portuguese by Lulu Bechtolsheim. MR**0219042** - A. Pełczyński,
*A generalisation of Stone’s theorem on approximation*, Bull. Acad. Polon. Sci. Cl. III.**5**(1957), 105–107, X (English, with Russian summary). MR**0086166** - William J. Pervin,
*Quasi-proximities for topological spaces*, Math. Ann.**150**(1963), 325–326. MR**151936**, DOI 10.1007/BF01470761
F. Riesz, - Yu. Smirnov,
*On proximity spaces in the sense of V. A. Efremovič*, Doklady Akad. Nauk SSSR (N.S.)**84**(1952), 895–898 (Russian). MR**0055660** - Yu. M. Smirnov,
*On proximity spaces*, Mat. Sbornik N.S.**31(73)**(1952), 543–574 (Russian). MR**0055661** - Eugene F. Steiner,
*The relation between quasi-proximities and topological spaces*, Math. Ann.**155**(1964), 194–195. MR**163278**, DOI 10.1007/BF01344159 - M. H. Stone,
*Boundedness properties in function-lattices*, Canad. J. Math.**1**(1949), 176–186. MR**29091**, DOI 10.4153/cjm-1949-016-5
H. Tong, - Hing Tong,
*Some characterizations of normal and perfectly normal spaces*, Duke Math. J.**19**(1952), 289–292. MR**50265**

*Quasi-proximities and order compactifications*, Notices Amer. Math. Soc.

**20**(1973), A-594. Abstract #73T-G122. A. Calder,

*Proximity algebras*, Carleton Math. Ser., no. 44, Carleton University, Ottawa, Canada.

*Stetigskeitsbegriff und abstrakte Mengenlehre*, Atti del IV Congresso Intern. dei Matem., Roma, 1908, v. 11, Rome, 1909, pp. 18-24.

*Some characterizations of normal and perfectly normal spaces*, Bull. Amer. Math. Soc.

**54**(1948), 65.

## Additional Information

- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**222**(1976), 65-96 - MSC: Primary 46E05; Secondary 54E05, 41A65
- DOI: https://doi.org/10.1090/S0002-9947-1976-0438094-0
- MathSciNet review: 0438094