The fit and flat components of barrelled spaces. (English) Zbl 0838.46002
Summary: The splitting theorem says that any given Hamel basis for a (Hausdorff) barrelled space \(E\) determines topologically complementary subspaces \(E_C\) and \(E_D\), and that \(E_C\) is flat, that is, contains no proper dense subspace. By use of this device it was shown that if \(E\) is non-flat it must contain a dense subspace of codimension at least \(\aleph_0\); here we maximally increase the estimate to \(\aleph_1\) under the assumption that the dominating cardinal \({\mathfrak d}\) equals \(\aleph_1\) [strictly weaker than the Continuum Hypothesis (CH)].
A related assumption strictly weaker than the generalized CH allows us to prove that \(E_D\) is fit, that is, contains a dense subspace whose codimension in \(E_D\) is \(\dim(E_D)\), the largest imaginable. Thus the two components are extreme opposites, and \(E\) itself is fit if and only if \(\dim(E_D)\geq \dim(E_C)\), in which case there is a choice of basis for which \(E_D= E\). Moreover, \(E\) is non-flat (if and) only if the codimension of \(E'\) is at least \(2^{\aleph_1}\) in \(E^*\).
These results ensure latitude in the search for certain subspaces of \(E^*\) transverse to \(E'\), as in the barrelled countable enlargement (BCE) problem, and show that every non-flat GM-space has a BCE.
A related assumption strictly weaker than the generalized CH allows us to prove that \(E_D\) is fit, that is, contains a dense subspace whose codimension in \(E_D\) is \(\dim(E_D)\), the largest imaginable. Thus the two components are extreme opposites, and \(E\) itself is fit if and only if \(\dim(E_D)\geq \dim(E_C)\), in which case there is a choice of basis for which \(E_D= E\). Moreover, \(E\) is non-flat (if and) only if the codimension of \(E'\) is at least \(2^{\aleph_1}\) in \(E^*\).
These results ensure latitude in the search for certain subspaces of \(E^*\) transverse to \(E'\), as in the barrelled countable enlargement (BCE) problem, and show that every non-flat GM-space has a BCE.
Keywords:
continuum hypothesis; Hamel basis; barrelled space; topologically complementary subspaces; barrelled countable enlargement; non-flat GM-spaceReferences:
| [1] | Robertson, Bull. Austral. Math. Soc. 37 pp 383– (1988) |
| [2] | DOI: 10.1007/BF01168742 · Zbl 0281.46005 · doi:10.1007/BF01168742 |
| [3] | van Douwen, Handbook of Set-theoretic Topology pp 111– (1984) · doi:10.1016/B978-0-444-86580-9.50006-9 |
| [4] | Tweddle, Topological vector spaces, algebras and related areas pp 3– (1994) |
| [5] | Robertson, Bull. Austral. Math. Soc. 22 pp 99– (1980) |
| [6] | Saxon, J. London Math. Soc. |
| [7] | DOI: 10.2307/2047662 · Zbl 0686.46002 · doi:10.2307/2047662 |
| [8] | Saxon, Topological vector spaces, algebras and related areas pp 24– (1994) |
| [9] | Saxon, J. London Math Soc. |
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