# compact sets

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• Oct 24th 2012, 12:53 AM
bskcase98
compact sets
before I write the problem I just want to clarify that this is not a graded assignment but a practice problem designed to help understand the topic. My professor has limited office hours so I am hoping someone can help me reason this out.

does the set Q intersection [0, 1] in R have an open cover that cannot be replaced by a finite sub-cover?

this is what I know:
(1) the question is asking if Q intersection [0, 1] is compact in R
(2) that [0, 1] is a compact interval
(3) that Q intersection [0, 1] is contained in [0,1] and that by the density of Q the Q intersection [0, 1] will be an open set

my initial thought is that no this intersection is not compact but I am not confident in that answer and I would really appreciate any assistance.

Thank you
• Oct 24th 2012, 01:46 AM
FernandoRevilla
Re: compact sets
Quote:

Originally Posted by bskcase98
I am not confident in that answer and I would really appreciate any assistance.

Denote $\displaystyle A=\mathbb{Q}\cap [0,1]$ then, $\displaystyle \bar{A}=[0,1]\neq A$, that is, $\displaystyle A$ is not a closed set, as a consequence, $\displaystyle A$ is not compact.