1

I want to show that the set $[a,b]$ is not compact in $\mathbb R$ with the lower limit topology. I tried as follows:

Since $[a,b]$ is uncountable, there exists a strictly increasing sequence $(x_n)\subset [a,b]$. Let $x=\sup\{x_n:n\in \mathbb N\}$. Since, $[a,b]$ is closed in $\mathbb R$ with the usual topology, $x\in [a,b]$. Now $[a,x_0),[x,b]$ and all sets of the form $[x_n,x_{n+1}),n\in \mathbb N$ form an open cover for $[a,b]$. This cover has no subcover as if we remove any of the sets $[x_n,x_{n+1})$ from the above collection, the union of the remaining subcollection can not contain $x_n$.

Am I right? Please rectify if I am wrong.

Anupam
  • 4,908

1 Answers1

2

Your cover works. However, the fact that $[a,b]$ is uncountable is irrelevant, and you can simplify matters by choosing a particular sequence: just let $$x_n=b-\frac{b-a}{2^{n+1}}$$ for $n\in\Bbb N$. This gives you a sequence converging to $b$, and you don’t need $[x,b)$ at all.

Brian M. Scott
  • 616,228