nLab
sigma-pretopological space
Redirected from "monoidal double categories".
Context
Topology
topology (point-set topology , point-free topology )
see also differential topology , algebraic topology , functional analysis and topological homotopy theory
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Basic statements
Theorems
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Contents
Idea
A pretopological space is a convergence space which has all intersections of convergent filters to an element also converge to that element. Thus, a σ \sigma -pretopological space should be a convergence space which should have countable intersections of convergent filters to an element converge to that element.
Definition
A σ \sigma -pretopological space S S is a set S S with a relation F → x F \to x from the set of filters on S S to S S itself, satisfying the following properties
Centred: The free ultrafilter at x x (the collection of all sets that x x belongs to) converges to x x :{ A | x ∈ A } → x . \{ A \;|\; x \in A \} \to x .
Isotone: If F F converges to x x and G G refines F F , then G G converges to x x :F → x ⇒ F ⊆ G ⇒ G → x . F \to x \;\Rightarrow\; F \subseteq G \;\Rightarrow\; G \to x .
Countably directed: The intersection of a countable family of filters that converge to x x itself converges to x x :∀ n ∈ ℕ . ( F n → x ) ⇒ ( ⋂ n ∈ ℕ F n ) → x . \forall n \in \mathbb{N}.(F_n \to x) \; \Rightarrow \; \left(\bigcap_{n \in \mathbb{N}} F_n\right) \to x .
Created on November 11, 2024 at 17:42:28.
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