Abstract
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 ≤ 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.
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McCluskey, A.E., McMaster, T.B.M. Realizing Quasiordered Sets by Subspaces of ‘Continuum-Like’ Spaces. Order 15, 143–149 (1998). https://doi.org/10.1023/A:1006100103089
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DOI: https://doi.org/10.1023/A:1006100103089