Homotopy Type Theory modulus of Cauchy convergence > history (Rev #2, changes)

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Definition

Let R T R T be a Archimedean directed ordered type integral domain , with let adenseSS be astrict orderTT , - and letR +R_{+}premetric space , be and the letsemiring?x:I→Sx:I \to S of be positive a terms inRRnet . with index typeII. A TT-modulus of Cauchy convergence is a function M:T→IM: T \to I with a type

Suppose SS is a R +R_{+}-premetric space, and let x:I→Sx:I \to S be a net with index type II. A R +R_{+}-modulus of Cauchy convergence is a function M:R +→IM: R_{+} \to I with a type

μ:∏ ϵ:T∏ i:I∏ j:I(i≥M(ϵ))×(j≥M(ϵ))→(x i∼ ϵx j)\mu:\prod_{\epsilon:T} \prod_{i:I} \prod_{j:I} (i \geq M(\epsilon)) \times (j \geq M(\epsilon)) \to (x_i \sim_{\epsilon} x_j)
μ:∏ ϵ:R +∏ i:I∏ j:I(i≥M(ϵ))×(j≥M(ϵ))→(x i∼ ϵx j)\mu:\prod_{\epsilon:R_{+}} \prod_{i:I} \prod_{j:I} (i \geq M(\epsilon)) \times (j \geq M(\epsilon)) \to (x_i \sim_{\epsilon} x_j)

The composition x∘Mx \circ M of a net xx and a TT-modulus of Cauchy convergence MM is also a net.

A R +R_{+}-Cauchy approximation is the composition M∘xM \circ x of a net xx and a R +R_{+}-modulus of Cauchy convergence MM.

See also

References

Revision on March 12, 2022 at 08:30:11 by Anonymous?. See the history of this page for a list of all contributions to it.

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