Prove that $[0,1]$ is not a compact subset of $\mathbb{R}$ with the lower limit topology, i.e. open sets are of the form $[a,b)$.
My question is will different topology affect compactness of a set? If this is so, why?
At first, when I see this question, I thought something is wrong with this question because I know that $[0,1]$ is compact by using Heine-Borel theorem.
Best Answer
In that case you can't use the Heine-Borel theorem because this theorem only apply to the case of normal topology.
To prove this proposition, you just find the open cover of $[0,1]$ such that every finite subcover does not cover $[0,1]$. Let $\mathcal{A}$ be a set of open sets defined as $$\mathcal{A}=\{[0,r):0<r<1\}\cup \{[1,2)\}$$ then $\mathcal{A}$ cover $[0,1]$. However every finite subcover of $\mathcal{A}$ does not cover $[0,1]$.