Summary
If there is a homeomorphic embedding of one set into another, the sets are said to be topologically comparable. Friedman and Hirst have shown that the topological comparability of countable closed subsets of the reals is equivalent to the subsystem of second order arithmetic denoted byATR 0. Here, this result is extended to countable closed locally compact subsets of arbitrary complete separable metric spaces. The extension uses an analogue of the one point compactification of ℝ.
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References
Alexandroff, P.: Über die Metrisation der im Kleinen kompakten topologischen Räume. Math. Ann.92, 294–301 (1924)
Brown, D.: Notions of closed subsets of a complete separable metric space in weak subsystems of second order arithmetic. In: Sieg, W. (ed.) Logic and computation. (Contemporary Math., vol. 106, pp. 39–50) Providence: AMS 1990
Brown, D., Simpson, S.: Which set existence axioms are needed to prove the Hahn-Banach theorem for separable Banach spaces? Ann. Pure Appl. Logic31, 132–144 (1986)
Friedmann, H.: Some systems of second order arithmetic and their use. In: James, R. (ed.) Proceedings of the International Congress of Mathematicians. (vol. 1, pp. 235–242) Vancouver: Canadian Mathematical Congress 1975
Friedman, H.: Systems of second order arithmetic with restricted induction (abstracts). J. Symb. Logic41, 557–559 (1976)
Friedman, H., Hirst, J.: Weak comparability of well orderings and reverse mathematics. Ann. Pure Appl. Logic47, 11–29 (1990)
Friedman, H., Hirst, J.: Reverse mathematics and homeomorphic embeddings. Ann. Pure Appl. Logic54, 229–253 (1991)
Kuratowski, K.: Topology. New York: Academic Press 1966
Simpson, S.: Which set existence axioms are needed to prove the Cauchy/Peano theorem for ordinary differential equations? J. Symb. Logic49, 783–802 (1984)
Simpson, S.: Subsystems ofZ 2 and reverse mathematics. In: Takeuti, G. (ed.) Proof theory. Amsterdam New York: North-Holland 1985
Simpson, S.: Subsystems of second order arithmetic (in preparation)
Thron, W.: Topological structures. New York: Holt, Rinehart, and Winston 1966
Urysohn, P.: Über die Metrisation der kompakten topologischen Räume. Math. Ann.92, 275–293 (1924)