Thread: Is this set thick in C0

Let C0 be the set of all continuous functions from [a,b] to R

Let A = { f C0: f(q) is irrational for all q Q} where Q is the set of rational numbers.

The question is: Is A a thick subset of C0.

EDIT: The answer I had here was very wrong

Originally Posted by southprkfan1

Let C0 be the set of all continuous functions from [a,b] to R

Let A = { f C0: f(q) is irrational for all q Q} where Q is the set of rational numbers.

The question is: Is A a thick subset of C0.


I said it is because if we let:

Gj = { f C0: f(q) Br(p) where r = 1/j and p is irrational for all q Q}

Then each Gj is open and dense and

A = {j=1 to ] Gj

so A is thick.

...thoughts?

What does thick mean?

Originally Posted by Drexel28

What does thick mean?

Right, I forgot some people use different terms. A set is thick in a metric space M if it the intersection of countably many open and dense sets in M. I believe it is also called a 2nd category set or something.