Homotopy Type Theory
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Definition
In Archimedean set ordered theory fields Let F S F S be an a Archimedean preconvergence ordered space field F S F S sequentially Hausdorff space if for all sequence s x : ∈ ℕ S ℕ → F x:\mathbb{N} x \to \in F S^\mathbb{N} , the type set of all limits of x x is has a at most one element proposition
isSequentiallyHausdorff ( F S ) ≔ ∏ x : ℕ → F ∀ isProp x ∈ S ℕ . { l ∈ S | isLimit S ( lim n → ∞ x , l ) } ≅ ∅ ∨ { l ∈ S | isLimit S ( n x , l ) ) } ≅ 𝟙 isSequentiallyHausdorff(F) isSequentiallyHausdorff(S) \coloneqq \prod_{x:\mathbb{N} \forall \to x F} \in isProp(\lim_{n S^\mathbb{N}. \to \{l \infty} \in x(n)) S \vert isLimit_S(x, l)\} \cong \emptyset \vee \{l \in S \vert isLimit_S(x, l)\} \cong \mathbb{1} All where Archimedean ordered fields are sequentially Hausdorff spaces.𝟙 \mathbb{1} is the singleton set.
Most general definition One could directly define the limit lim n → ∞ ( − ) ( n ) \lim_{n \to \infty} (-)(n) partial function
Let S S predicate → \to S S ℕ → S \mathbb{N} \to S S S S S sequentially Hausdorff space if for all sequence s x : ℕ → S x:\mathbb{N} \to S limits of x x proposition
lim n → ∞ ( − ) ( n ) ∈ ( ℕ → S ) → 𝒰 S \lim_{n \to \infty} (-)(n) \in (\mathbb{N} \to S) \to_{\mathcal{U}} S isSequentiallyHausdorff ( S ) ≔ ∏ x : ℕ → S isProp ( ∑ l : S x → l ) isSequentiallyHausdorff(S) \coloneqq \prod_{x:\mathbb{N} \to S} isProp\left(\sum_{l:S} x \to l\right) In homotopy type theory One Let could directly define the limitlim n → ∞ S ( − ) ( n ) \lim_{n S \to \infty} (-)(n) as be a partial preconvergence function space . Then S S is a sequentially Hausdorff space if for all sequence s x : ℕ → S x:\mathbb{N} \to S , the type of all limits of x x is a proposition
lim n → ∞ isSequentiallyHausdorff ( − ) ( n ) : ( ℕ → S ) → 𝒰 ≔ S ∏ x : ℕ → S isProp ( ∑ l : S x → l ) \lim_{n isSequentiallyHausdorff(S) \coloneqq \prod_{x:\mathbb{N} \to \infty} S} (-)(n): isProp\left(\sum_{l:S} (\mathbb{N} x \to S) l\right) \to_{\mathcal{U}} SOne could directly define the limit lim n → ∞ ( − ) ( n ) \lim_{n \to \infty} (-)(n) partial function
lim n → ∞ ( − ) ( n ) : ( ℕ → S ) → 𝒰 S \lim_{n \to \infty} (-)(n): (\mathbb{N} \to S) \to_{\mathcal{U}} S
Properties
Every Hausdorff space and thus every Archimedean ordered field is a sequentially Hausdorff space.
See also
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