Is Cantor set is compact with respect to lower limit Topology on $R$?

Question

Is Cantor set is compact with respect to lower limit Topology on $R$? I know Cantor set is compact with respect to usual topology. But I think it's not compact in lower limit Topology.. Because I think we can write Cantor set as [0,$a_1$) union [$a_1$,$a_2$)...union 1. But I am not able to write it explicitly.. I am not able to find it's cover that will have infinite subcover. Please give me some hint..

Answer 1

I assume $R$ refers to $\mathbb{R}$, i.e., the set of real numbers. The answer is no, the Cantor set is not compact with respect to the lower limit topology $\tau$. This follows from the fact that every compact subset of $(\mathbb{R},\tau)$ must be countable, while the Cantor set itself is uncountable.

Answer 2

In the Sorgenfrey Line (also called the lower-limit topology), the family $\{[0,x): 0<x<1\}\cup \{[1,2)\}$ is an open cover of the Cantor set with no finite sub-cover