Products of Steiner’s quasi-proximity spaces
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- by E. Hayashi PDF
- Proc. Amer. Math. Soc. 51 (1975), 225-230 Request permission
Abstract:
E. F. Steiner introduced a quasi-proximity $\delta$ satisfying $A\delta B \operatorname {iff}\; \{ x\} \delta B$ for some $x$ of $A$. The purpose of this paper is to describe the Tychonoff product of topologies in terms of Steiner’s quasi-proximities. Whenever $({X_a},{\delta _a})$ is the Steiner quasi-proximity space, the product proximity on $X = \Pi {X_a}$ can be given, by using the concept of finite coverings, as the smallest proximity on $X$ which makes each projection $\delta$-continuous.References
- S. A. Naimpally and B. D. Warrack, Proximity spaces, Cambridge Tracts in Mathematics and Mathematical Physics, No. 59, Cambridge University Press, London-New York, 1970. MR 0278261
- Eugene F. Steiner, The relation between quasi-proximities and topological spaces, Math. Ann. 155 (1964), 194–195. MR 163278, DOI 10.1007/BF01344159
- Frederick W. Stevenson, Product proximities, Fund. Math. 76 (1972), no. 2, 157–166. MR 305361, DOI 10.4064/fm-76-2-157-166
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 51 (1975), 225-230
- MSC: Primary 54E05
- DOI: https://doi.org/10.1090/S0002-9939-1975-0370512-3
- MathSciNet review: 0370512