In mathematical set theory, Baumgartner's axiom (BA) can be one of three different axioms introduced by James Earl Baumgartner.
This would be true for the real line itself under the continuum hypothesis.
-dense subsets of the real line are order-isomorphic, providing a higher-cardinality analogue of Cantor's isomorphism theorem that countable dense subsets are isomorphic.
It is consistent with a combination of ZFC, Martin's axiom, and the negation of the continuum hypothesis,[1] but not implied by those hypotheses.
[2] Another axiom introduced by Baumgartner (1975) states that Martin's axiom for partially ordered sets MAP(κ) is true for all partially ordered sets P that are countable closed, well met and ℵ1-linked and all cardinals κ less than 2ℵ1.