Confusion on the definition of the standard topology on $\mathbb{R}$ So I've been learning a bit about topology recently so I am a complete novice and there are a lot of definitions that confuse me my first confusion regards the topology on $\mathbb{R}$. I think that the standard  topology consists of all open intervals $(a,b)$ and that the topology is generated by the basis that consists of all open balls.
However open balls are used only in metric spaces since we have a distance function so when using these open balls do we no longer have a topological space but a metric space?
My second confusion is that of half open intervals in the standard topology on $\mathbb{R}$ for example I know that singletons are closed when considered as a subset of the reals such as $[0]$ is closed however what is $[0,2)$ considered?
Thanks in advance.

A metric space is a topological space (the converse is not always true). The set $ [0,2) $ is neither open nor closed. So are the set $ (0,1) \cup [4,5] $ or the set of rationals $ \mathbb{Q} $.
