I had this question on my final:
Let $(X,d)$ be a metric space. Prove that the set of points where $X$ is locally connected is the countable intersection of open subsets of $X$.
I wasn't quite sure how to approach the problem; I know that the components of open subsets of $X$ are open if $X$ itself is locally connected, but I wasn't sure where to go with the information (if anywhere at all). Any hints? If possible, I'd prefer hints to begin with so I can work out a solution. After I have solved it, anyone can return to edit their solution to be more complete so as not to detract from the value of MSE as a reference site.