Compact subspace of $\mathbb{R}$ with lower limit topology must be countable.

Question

Any compact subset of $\mathbb{R}_{l} $ must be a countable set.

Consider the open cover $\{[n,n+1): n \in \Bbb Z\}$ of $\Bbb R $ which has no subcover. So $\Bbb R $ is not compact with respect to lower limit (or Sorgenfrey) topology.

But how to answer this question?

Answer 1

HINT: Supppose that $X\subseteq\Bbb R_\ell$ is uncountable; you want to show that $X$ is not compact. Show that there are an $x\in X$ and a strictly increasing sequence $\langle x_n:n\in\Bbb N\rangle$ in $X$ that converges to $x$ in the usual topology. Then consider a countable open cover of $X$ that includes sets of the form $[x_n,x_{n+1})$, among others.

Added: Let $A$ be the set of points of $X$ that are not the limit of a strictly increasing sequence in $X$.

• Show that for each $x\in A$ there is an $\epsilon_x>0$ such that $(x-\epsilon_x,x)\cap X=\varnothing$.
• Explain why $\{(x-\epsilon_x,x):x\in A\}$ is a pairwise disjoint family of non-empty open intervals.
• Deduce that $A$ is countable and hence that $X\setminus A\ne\varnothing$.