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Bruno, Jorge; McCluskey, Aisling

doi:10.1016/j.topol.2012.05.014

Topology and its Applications

2012

DOI: 10.1016/j.topol.2012.05.014

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Topologies on X as points within 2P(X)

Jorge Bruno

Aisling McCluskey

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2013

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Topologies as points within a Stone space: Lattice theory meets topology

Bruno

McCluskey

2013

Topology and its Applications

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For a non-empty set X, the collection T op(X) of all topologies on X sits inside the Boolean lattice P(P(X)) (when ordered by set-theoretic inclusion) which in turn can be naturally identified with the Stone space 2 P(X) . Via this identification then, T op(X) naturally inherits the subspace topology from 2 P(X) . Extending ideas of Frink (1942), we apply lattice-theoretic methods to establish an equivalence between the topological closures of sublattices of 2 P(X) and their (completely distributive) completions. We exploit this equivalence when searching for countably infinite compact subsets within T op(X) and in crystalizing the Borel complexity of T op(X). We exhibit infinite compact subsets of T op(X) including, in particular, copies of the Stone-Čech and one-point compactifications of discrete spaces.

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Topologies as points within a Stone space: Lattice theory meets topology

Bruno

McCluskey

2013

Topology and its Applications

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Topologies on X as points within 2P(X)

Cited by 1 publication

References 6 publications

Topologies as points within a Stone space: Lattice theory meets topology

Topologies as points within a Stone space: Lattice theory meets topology

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