Consider $X = [0,1]$ and $A = (a,b)$ (where $0 < a < b < 1$).
I want to prove $X/A$ is connected and compact. Here's my approach :
If I take an open cover of $X/A$, the corresponding cover in X will admit a finite subcover since X is compact (w.r.t. the euclidean topology), so the original subcover will also admit a finite subcover. This proves compacity. Then for connectedness I just use the fact that the quotient map is continuous and surjective so its image is connected.
Then I am asked if the space $X/A$ is Hausdorff.
I can see that it is not, because there exist only one point between $a$ and $b$ (which is the point to which the whole set $(a,b)$ is sent to by the quotient map), so any open set containing either $a$ or $b$ will contain this point so their intersection cannot be empty.
Is this the right intuition ? If so how would I formalize it ?