English
The collection of topologies on α forms a complete lattice, with ⊥ the discrete topology and ⊤ the indiscrete topology. The infimum of any family is the topology generated by the union of their open sets, while the supremum is the topology whose open sets are those open in every member.
Русский
Множество топологий на α образует законченную решетку; нижняя граница — дискретная топология, верхняя — абсурдная. Первообразование: пересечение открытых множеств даёт инфimum; пересечение топологий даёт supremum.
LaTeX
$$$\\text{Topologies on } \\alpha \\text{ form a complete lattice; }\\bot = \\text{DiscreteTopology}(\\alpha),\\; \\top = \\text{IndiscreteTopology}(\\alpha).\\;\\inf\\mathcal{T} = \\mathrm{generateFrom}\\bigcup_{t\\in\\mathcal{T}}\\{ U: U\\subseteq\\alpha \\mid \\IsOpen_t(U)\\},\\; \\sup\\mathcal{T} = \\{ U \\subseteq\\alpha \\mid \\forall t\\in\\mathcal{T}, \\IsOpen_t(U)\\}.$$$
Lean4
/-- Topologies on `α` form a complete lattice, with `⊥` the discrete topology
and `⊤` the indiscrete topology. The infimum of a collection of topologies
is the topology generated by all their open sets, while the supremum is the
topology whose open sets are those sets open in every member of the collection. -/
instance : CompleteLattice (TopologicalSpace α) :=
(gciGenerateFrom α).liftCompleteLattice
