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On Disjoint Sums in the Lattice of Linear Topologies

Abstract

Let M be a vector space over a skew-field equipped with the discrete topology, \(\mathcal{L}\)(M) be the lattice of all linear topologies on M ordered by inclusion,and τ*, τ0, τ1\(\mathcal{L}\)(M). We write τ1 = τ* ⊔ τ0 or say that τ1 is a disjoint sum of τ* and τ0 if τ1 = inf{τ0, τ*} and sup{τ0, τ*} is the discrete topology. Given τ1, τ0\(\mathcal{L}\)(M), we say that τ0 is a disjoint summand of τ1 if τ1 = τ* ⊔ τ0 for a certain τ*\(\mathcal{L}\)(M). Some necessary and some sufficient conditions are proved for τ0 to be a disjoint summand of τ1.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 3–18, 2003.

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Arnautov, V.I., Filippov, K.M. On Disjoint Sums in the Lattice of Linear Topologies. J Math Sci 128, 3335–3344 (2005). https://doi.org/10.1007/s10958-005-0270-4

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Keywords

  • Vector Space
  • Discrete Topology
  • Linear Topology