ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. 62,   2   (1993)
pp.   197-205

A HAMILTONIAN PROPERTY OF CONNECTED SETS IN THE ALTERNATIVE SET THEORY
P. ZLATOS

Abstract.  The representation of indiscernibility phenomena by $\pi$-equivalences and of accessibility phenomena by $\sigma$-equivalences enables a graph-theoretical formulation of topological notions in the alternative set theory. Generalizing the notion of Hamiltonian graph we will introduce the notion of Hamiltonian embedding and prove that for any finite graph without isolated vertices there is a Hamiltonian embedding into any infinite set connected with respect to some $\pi$- or $\sigma$-equivalence. Roughly speaking, in some sense this means that each such an infinite connected set, (in particular, each connected set in a complete metrizable topological space), contains each finite graph inside, and even is exhausted by the images of its edges. Moreover, the main Theorem 3, dealing with the so called deeply connected sets, is in fact a theorem of nonstandard arithmetic.

AMS subject classification.  03E70, 05C45, 54J05; Secondary 03H05, 03H15, 05C10, 54D05
Keywords.  Alternative set theory, topology, $\pi$-equivalence, $\sigma$-equivalence, graph, path, connected, deeply connected, Hamiltonian circle, nonstandard arithmetic