Showing every open cover of [0,1] has a finite sub-cover with connectedness. Question is as in the title. We are not allowed to invoke compactness.
Professor provides this argument: Suppose $F=\{O_\alpha|\alpha \in I\}$ is an open cover of $[0,1]$. Define $J\subseteq [0,1]$ such that $J$ can be covered by some finite subset of $F$. Then $J$ is open because for any point $a\in J$, $a$ is in some $O_1$, and $O_1\cap J \subseteq J$. Then $a$ is an interior point of $J$. I don't understand why it is interior. I know there is an open interval around $a$ that is in $O_1$.
Also $J$ is closed, because any cluster point $x_0$ of $J$ in $[0,1]$ is contained in some $O_\beta$. Then $x_0 \in J$. I don't see why $x_0$ has to be in $J$ at all, $O_\beta$ could have non-trivial intersection with the finite sub-cover of $J$ but not be a member of the sub-cover.
In fact I think the above argument is broken. Any ideas to make a cogent argument out of this?
 A: Are you sure that you’ve accurately reported the definition of $J$? The argument works if
$$J=\{x\in[0,1]:[0,x]\text{ can be covered by a finite }F_x\subseteq F\}\,.$$
Let $x\in J$. Then there are an $O\in F_x$ such that $x\in O$ and an $\epsilon>0$ such that $(x-\epsilon,x+\epsilon)\subseteq O$; clearly $[0,x+\epsilon)\subseteq\bigcup F_x$, so $\left[0,x+\frac{\epsilon}2\right]\subseteq J$, and $\left(0,x+\frac{\epsilon}2\right)$ is an open nbhd of $x$ contained in $J$. This shows that $J$ is open in $[0,1]$.
Now let $x$ be any cluster point of $J$ in $[0,1]$. There are an $O\in F$ and an $\epsilon>0$ such that $(x-\epsilon,x+\epsilon)\subseteq O$, and since $x$ is a cluster point of $J$, there is a $y\in(x-\epsilon,x+\epsilon)\cap J$. Let $F_y$ be a finite subset of $F$ covering $[0,y]$; then $F_y\cup\{O\}$ is a finite subset of $F$ covering $[0,y]\cup(x-\epsilon,x+\epsilon)$. In particular, $F_y\cup\{O\}$ covers $[0,x]$, so $x\in J$. This shows that $J$ is closed in $[0,1]$.
Clearly $0\in J$, so $J\ne\varnothing$, and $[0,1]$ is connected, so $J=[0,1]$, $1\in J$, and there is a finite $F_1\subseteq F$ covering $[0,1]$.
