The background of my question is $(\mathbb{R},\mathcal{F})$, the real line with $\mathcal{F}$ a topology defined as $$\mathcal{F}:=\{\emptyset, \{[x,t) : t > x\}\}.$$
Now, how is the prove to get $[0,1]$ is nos compact? If it is possible, I´d like to see a simple prove (that is, not using sequences).
For example: $\{[0,\frac{1}{2}),[\frac{1}{2},2)\}$ is a covering of $[0,1]$ such that not admitts a subcovering?
Thank you very much, and kind regards!
Remark that $\{1\}$ is open in $[0,1]$ for the induced topology since $\{1\}=[1,2)\cap [0,1]$. Consider $[0,1-1/n)\cup\{1\}$ its a covering which does not has a finite subcovering.