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Boundedness in a Quasi-Uniform Space

  • M. G. Murdeshwar (a1) and K. K. Theckedath (a2)

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Although a nontopological concept, boundedness seems to be of considerable importance in a topological space. 'There are many topological problems in which it is essential to be able to make this distinction' (between bounded and unbounded sets) [1]. Boundedness and in particular boundedness-preserving' uniform spaces appear to have applications to topological dynamics [4].

In spite of this importance, there have been only isolated attempts at developing the concept. Alexander [1] and Hu [7] tried the axiomatic approach. Hu, for example, calls a nonempty family of sets a boundedness if is hereditary and closed under finite union.

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References

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1. Alexander, J. W., On the concept of a topological space, Proc. Nat. Acad. Sci. U.S.A. 25 (1939), 52-54.
2. Bourbaki, N., Topologie générale, Hermann, Paris, 1950.
3. Bourbaki, N., Topologie générale, 4th éd., Hermann, Paris, 1965.
4. Bushaw, D., On boundedness in uniform spaces, Fund. Math. 56 (1965), 295-300.
5. Császár, A., Fondements de la topologie générale, Gauthier-Villars, Paris, 1960.
6. Hejcman, Jan, On Conservative uniform spaces, Comment. Math. Univ. Carolinae 7 (1966), 411-417.
7. Hu, S. T., Boundedness in a topological space, J. Math. Pures Appl. 28 (1949), 287-320.
8. Murdeshwar, M. G., and Naimpally, S. A., Quasi-uniform topological spaces, Noordhoff Groningen, 1966.
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Boundedness in a Quasi-Uniform Space

  • M. G. Murdeshwar (a1) and K. K. Theckedath (a2)

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