Stream: general

Topic: notation in topological space file

Kevin Buzzard (Mar 12 2018 at 18:51):

In analysis/topology/topological_space.lean we have s and t being used for more than one thing. Even in the definition of a topological space we have

(is_open_inter  : ∀s t, is_open s → is_open t → is_open (s ∩ t))
(is_open_sUnion : ∀s, (∀t∈s, is_open t) → is_open (⋃₀ s))


Here s is an open set on one line, and a set of open sets on the next. Is this sort of thing regarded as OK? We're doing Xena tonight (it's usually Thursdays but I'm busy this Thurs) and a 2nd year undergraduate who has just learnt what a topological space is, is trying to read this mathlib file and this sort of notational trickery is not helping. Would @Mario Carneiro be interested in a PR which only contained changes of the form

(is_open_inter  : ∀s₁ s₂, is_open s₁ → is_open s₂ → is_open (s₁ ∩ s₂))
(is_open_sUnion : ∀I, (∀s∈I, is_open s) → is_open (⋃₀ I))


? For me, that is more readable, but might not conform to some sort of mathlib style (I'm not sure about this). Later on in is_open_sUnion we have a t in the statement and a different t in the proof. Of course none of this is logically wrong, but it does strike me as a strange design decision in some sense. Maybe mathematicians don't label their theorems as well as computer scientists might want them to, but I think they label their objects in a more consistent manner than this (e.g it would be considered bad writing to have s representing more than one thing, particularly two different types in consecutive sentences -- although of course I've seen it happen!)

Mario Carneiro (Mar 12 2018 at 18:53):

I often use capital letters for higher order sets. So is_open_sUnion would become ∀S, (∀s∈S, is_open s) → is_open (⋃₀ S)

Mario Carneiro (Mar 12 2018 at 18:53):

I believe there is a similar convention of changing font registers in standard math

Mario Carneiro (Mar 12 2018 at 18:54):

I usually reserve I or ι for the index set of a type/set family, say if I was discussing the indexed union instead of the set union

Last updated: May 12 2021 at 05:19 UTC