# Zulip Chat Archive

## Stream: maths

### Topic: computers are smart??

#### Reid Barton (Jun 04 2018 at 15:08):

So I was going along constructing coequalizers in Top, like you do. A coequalizer is a kind of quotient, so I needed to prove that if $Z$ is the quotient of $Y$ by a relation, then the quotient map $Y \to Z$ is continuous, so that it's a morphism in Top.

I was all set to add the lemma for that to my `continuity`

tactic, when I noticed the tactic had already succeeded before I did so! What happened?

#### Reid Barton (Jun 04 2018 at 15:09):

It turns out that continuity of the quotient map $Y \to Z$ is *definitionally equivalent* to continuity of the identity map on $Z$, because a set is defined to be open in $Z$ if and only if its preimage is open in $Y$. So the tactic found that it could just apply `continuous_id`

!

#### Johan Commelin (Jun 04 2018 at 17:08):

Wunderbar!

#### Johan Commelin (Jun 05 2018 at 07:42):

In fact, Reid, how did you define the topology on $Z$? Because there is a bunch of stuff on coinduced topologies in mathlib, and that would also give you continuity of the quotient map by definition.

#### Reid Barton (Jun 05 2018 at 11:52):

I did use the coinduced topology (see PR at https://github.com/leanprover/mathlib/pull/155/commits/b60f3687e8692f118c385f958d48d31388593298#diff-1c17754b46d709d0b8e22318f94035cdR903)

#### Johan Commelin (Jun 05 2018 at 11:53):

I see. So your `continuity`

tactic is not in that PR?

Last updated: May 12 2021 at 08:14 UTC