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Mad families of vector subspaces and the smallest nonmeager set of reals

Iian B. Smythe

Abstract

We show that a parametrized $\diamondsuit$ principle, corresponding to the uniformity of the meager ideal, implies that the minimum cardinality of an infinite maximal almost disjoint family of block subspaces in a countable vector space is $\aleph_1$. Consequently, this cardinal invariant is $\aleph_1$ in the Miller model.

Mad families of vector subspaces and the smallest nonmeager set of reals

Abstract

We show that a parametrized principle, corresponding to the uniformity of the meager ideal, implies that the minimum cardinality of an infinite maximal almost disjoint family of block subspaces in a countable vector space is . Consequently, this cardinal invariant is in the Miller model.

Paper Structure

This paper contains 3 sections, 10 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Cardinal invariants and parametrized $\diamondsuit$ principles
  3. The result

Key Result

Theorem 1.1

$\mathrm{non}(\mathcal{M})\leq\mathfrak{a}_{\mathrm{vec},F}$.

Theorems & Definitions (13)

  • Theorem 1.1: Corollary 2.11 in Smythe_mad_vec
  • Theorem 2.1: cf. Theorem 2.4 in MR917147
  • Theorem 2.2: Theorem 6.6 in MR2048518
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5: cf. Theorem 7.3.46 in MR1350295
  • Corollary 2.6
  • Theorem 3.1
  • Lemma 3.2
  • ...and 3 more