Let $A=[0,1]$. Show that $A$ is neither compact nor connected in the Sorgenfrey line, $\tau_{[,)}$, and that there is no neighborhood of $0$ compact.
For the connectedness part, I thought that $[0,1)$ might work. It is open in $\tau_{[,)}$, and $$\mathbb{R}\setminus [0,1)=(-\infty, 0)\cup [1,\infty)= \bigcup_{a\in \mathbb{R}^+} [-a,0) \cup \bigcup_{b\in \mathbb{R}^+}[1,b)$$ which is a union of open sets and therefore $[0,1)$ is also closed. Since there exists a non trivial subset which is both open and closed, then $(\mathbb{R},\tau_{[,)})$ is not connected, and since $[0,1)\subset A$, then A is not connected neither.
For the compactness part, I'm still struggling with the open cover/ finite subcover definition, so I propose as an attempt the following open cover: $$\mathfrak{A}=\left\{ [ 0,1+1/n) \mid n\in \mathbb{Z}^+ \right\}$$ which I believe it doesn't have a finite subcover, but I lack of a formal proof of this fact.
The last paragraph would also provide a proof that there is not a compact neighborhood of $0$. Let $[a,b)$ be such a neighborhood, then $$\left\{ [a,b+1/n) \mid n\in \mathbb{Z}^+ \right\}$$ would also be an open cover of $[a,b)$ that doesn't have a finite subcover, in contradiction with the hypothesis that $[a,b)$ is compact.
Is this proof correct? Thanks in advance!