Metric Baumgartner theorems and universality

It is consistent with the axioms of set theory that for every metric space $X$ which is isometric to some separable Banach space or to Urysohn’s universal separable metric space $\mathbb U$ the following holds: \begin{quote} $ {(\star)_X}$ \emph{There exists a nowhere meager subspace of $X$ of cardinality $\aleph_1$ and any two nowhere meager subsets of $X$ of cardinality $\aleph_1$ are almost isometric to each other.} \end{quote} As a corollary, it is consistent that the Continuum Hypothesis fails and the following hold: \begin{enumerate} \item There exists an almost-isometry ultrahomogeneous and universal element in the class of separable metric spaces of size $\aleph_1$. \item For every separable Banach space $X$ there exists an almost-isometry conditionally ultrahomogeneous and universal element in the class of subspaces of $X$ of size $\aleph_1$. \item For every finite dimensional Banach space $X$, there is a \emph{unique} universal element up to almost-isometry in the class of subspaces of $X$ of size $\aleph_1$ \end{enumerate}

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Published 1 January 2007