Mathematics Department, NUI, Galway

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NUI Galway Mathematics
Preprint IRL-GLWY-2006-002 (PDF)

$℘(ℝ)$ , ordered by homeomorphic embeddability, does not represent all posets of cardinality $2$ ^c.

Robin W. Knight and Aisling McCluskey

Abstract

We prove it to be consistent that there is a poset of cardinality $2$ ^c which is not realizable in $℘(ℝ)$ , ordered by homeomorphic embeddability. This addresses and answers resolutely (and in the negative) the open question of whether there is a ZFC theorem that all posets of cardinality $2$ ^c can be represented by subspaces of the real line ordered by homeomorphic embeddability. This question arises from the pioneering work of Banach, Kuratowski and Sierpinski in the area and this result complements the recent work of [9], thus providing a proof of independence.

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School of Mathematics, Statistics and Applied Mathematics
National University of Ireland, Galway, University Road, Galway, Ireland.
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This page was last updated Monday, May 10

NUI Galway Mathematics Preprint IRL-GLWY-2006-002 (PDF)

℘(ℝ), ordered by homeomorphic embeddability, does not represent all posets of cardinality 2c.

Abstract

NUI Galway Mathematics
Preprint IRL-GLWY-2006-002 (PDF)

$℘(ℝ)$ , ordered by homeomorphic embeddability, does not represent all posets of cardinality $2$ ^c.