An embedding theorem for certain spaces with an equidistant property

Nadler, Sam B.

doi:10.1090/S0002-9939-1976-0410686-X

An embedding theorem for certain spaces with an equidistant property
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by Sam B. Nadler

Proc. Amer. Math. Soc. 59 (1976), 179-183

DOI: https://doi.org/10.1090/S0002-9939-1976-0410686-X

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Abstract:

It is shown that certain metric spaces with the unique equidistant property can be topologically embedded in the real line. Several examples are given which show that the spaces considered are nontrivial, and which indicate that the technique of proof is necessary.

References

Anthony D. Berard Jr., Characterizations of metric spaces by the use of their midsets: Intervals, Fund. Math. 73 (1971/72), no. 1, 1–7. MR 295300, DOI 10.4064/fm-73-1-1-7
A. D. Berard Jr. and W. Nitka, A new definition of the circle by the use of bisectors, Fund. Math. 85 (1974), no. 1, 49–55. MR 355991, DOI 10.4064/fm-85-1-49-55
Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
K. Kuratowski, S. B. Nadler Jr., and G. S. Young, Continuous selections on locally compact separable metric spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 18 (1970), 5–11 (English, with Russian summary). MR 264630
A. Lelek and W. Nitka, On convex metric spaces. I, Fund. Math. 49 (1960/61), 183–204. MR 124882, DOI 10.4064/fm-49-2-183-204
L. D. Loveland and J. E. Valentine, Characterizing a circle with the double midset property, Proc. Amer. Math. Soc. 53 (1975), no. 2, 443–444. MR 388242, DOI 10.1090/S0002-9939-1975-0388242-0
L. D. Loveland and J. E. Valentine, Convex metric spaces with $0$-dimensional midsets, Proc. Amer. Math. Soc. 37 (1973), 568–571. MR 310817, DOI 10.1090/S0002-9939-1973-0310817-3

Generalized midset properties characterize geodesic circles and intervals

L. D. Loveland and S. G. Wayment, Characterizing a curve with the double midset property, Amer. Math. Monthly 81 (1974), 1003–1006. MR 418059, DOI 10.2307/2319308
Edwin W. Miller, On Subsets of a Continuous Curve which Lie on an Arc of the Continuous Curve, Amer. J. Math. 54 (1932), no. 2, 397–416. MR 1506905, DOI 10.2307/2371004