Communities

Writing
Writing
Codidact Meta
Codidact Meta
The Great Outdoors
The Great Outdoors
Photography & Video
Photography & Video
Scientific Speculation
Scientific Speculation
Cooking
Cooking
Electrical Engineering
Electrical Engineering
Judaism
Judaism
Languages & Linguistics
Languages & Linguistics
Software Development
Software Development
Mathematics
Mathematics
Christianity
Christianity
Code Golf
Code Golf
Music
Music
Physics
Physics
Linux Systems
Linux Systems
Power Users
Power Users
Tabletop RPGs
Tabletop RPGs
Community Proposals
Community Proposals
tag:snake search within a tag
answers:0 unanswered questions
user:xxxx search by author id
score:0.5 posts with 0.5+ score
"snake oil" exact phrase
votes:4 posts with 4+ votes
created:<1w created < 1 week ago
post_type:xxxx type of post
Search help
Notifications
Mark all as read See all your notifications »
Q&A

"Pointwise equicontinuity implies uniform" implies compact

+3
−0

Suppose $(M,d)$ is a metric space such that every sequence $f_n:M\to \Bbb R$ which is pointwise equicontinuous is also uniformly equicontinuous. Does this imply that $M$ is a compact metric space?


My thoughts: If true, we could try to show compactness directly or show sequential compactness and totally bounded.

Starting with the first idea, we take any open cover of $M$. One would think that we want to use this to construct some pointwise equicontinuous sequence of functions. No good ideas come to my mind.

Ok, sequential compactness, let ${x_n}$ be a Cauchy sequence in $M$. Maybe we take $f_n(x) = x_n$ the constant function, but that's dumb because it's already uniformly equicontinuous. I can't think of any interesting pointwise equicontinuous functions.


If false we want to construct a counter-example, which means finding a non-compact domain where pointwise equicontinuity implies uniform equicontinuity. My first thought is to try to take the domain to be some subset of the reals which is either not closed or not bounded. Probably start by trying "not bounded", like maybe the domain is $\Bbb N$. Is every pointwise equicontinuous sequence of functions uniformly equicontinuous? It doesn't seem like it should be--having an infinity of points seems like room enough to make a pointwise equicontinuous sequence which is not uniformly so.

I think any domain of reals containing an open interval will not have the property that p.w.e.c implies u.e.c., so maybe I should stop looking for examples in the reals.

What happens with the discrete metric on some set? The only compact sets are the finite sets, so if I can find any infinite domain where p.w.e.c. implies u.e.c. then I'm done. But if the natural numbers didn't work, then this probably can't work either.

History
Why does this post require moderator attention?
You might want to add some details to your flag.
Why should this post be closed?

0 comment threads

1 answer

+0
−0

Oh wait never mind. The natural number example works because every family of functions is equicontinuous, because delta=1 works for everything.

History
Why does this post require moderator attention?
You might want to add some details to your flag.

0 comment threads

Sign up to answer this question »