# empty interior

• Sep 23rd 2011, 12:48 PM
Kat-M
empty interior
Show that a compact set in an infinite dimensional space has an empty interior.

i know that a normed space is finite dimensional iff a closed unit ball is compact. but i am not sure how to make use of this theorem.

Let K be a compact set of X, which is infinite dimensional.
Assume that int(K) is not empty.
suppose x is in K. then there is an open ball with radius r, centered x. and this ball is entirely contained in K.

how can i use the compact closed unit ball to make a contradiction? help please.
• Sep 23rd 2011, 01:30 PM
girdav
Re: empty interior
You can choose r such that the closed ball of radius r centered at x is contained in K. Therefore, this ball is compact. But as you noticed, we can't find any nonempty compact ball (since we can write a non empty ball as f(B), where B is the unit closed ball and f a continuous map).
• Sep 23rd 2011, 02:33 PM
Kat-M
Re: empty interior
So since i have an open ball with radius r, centered at x, contained K, can i choose 0<r'<r and find a closed ball with radius r', centered at x?
this closed ball is entirely contained in K.
but this is not enough to say that the dimension of X is finite, is it? this seems too easy. and i did not use the theorem properly.