I have a problem that is probably relatively easy, I'm just a little lost since I've just started our chapter on connectedness. Here's the problem:
If is a totally disconnected, compact space, prove that has a clopen basis.
Our professor gave us a hint: For each , let be an open set containing and work with the complement to find a clopen such that .
I guess my problem is that I'm not seeing what a clopen set in this context looks like, and what compactness has to do with this. Any help to get me started is appreciated.