Given an ordered set E and a topological space X, we say that E can be realized within X if there is an injection j from E into the class of (homeomorphism classes of) subspaces of X such that, for $x,y$ in E, x \le y if and only if j(x) is homeomorphically embeddable into j(y). It is known, for instance, that transfinite induction demonstrates that every partially-ordered set of cardinality c (and some larger ones) can be realized within the real line. We explore aspects of the realizability problem, indicating in particular how to weaken the hypothesis on E from partial- to quasi-order, and seeking to isolate the characteristics of the real line that are relevant here.

key words and phrases: quasiorder, ordering by embeddability, transfinite induction, periodic sets, compressibility.

AMS (MOS) Subject Classification: 06A06, 54H10.

This article has been published. A. E. McCluskey\ and\ T. B. M. McMaster, Realizing quasiordered sets by subspaces of "continuum-like" spaces. Order {\bf 15} (1998/99), no.~2, 143--149; MR 2001i:54042; Zbl 0951.06003