Is it guaranteed that any topological space would always have an open cover? I think it should, but I wanted to check why. I feel like it's maybe related to the base of a topology? I know the base elements cover X, but they aren't open themselves correct? My other idea was that every topological space is itself open, so it's covered by itself at least. So is there always at least one open cover (the space itself) and then other open covers can exist depending on the specific space? Or are there other general open covers that exist for all topological spaces?
Sorry if it sounds like I answered my own question, but I'm just not confident about any of this and would appreciate someone dissecting my thoughts and presenting it more clearly.
Every nonempty topological space $X$ has two open covers namely $\{X\}$ and $\{X,\varnothing\}$. And this is the best one can find since the only two open covers of an indiscrete space are these.