I've been reading through Schaum's Outlines General Topology and I'm currently looking at regular spaces. The Schaum's books cites the following definition and example:
After having read all that, note that the example says that $\{b\}$ is not closed. This must mean that $\{b\}^c$ is not open. Well, granted, it is true that $\{b\}^c=\{a,c\}$ which is not in our topology $\mathcal{T}$. Therefore, $\{a,c\}$ is not open, and thus $\{b\}$ is not closed. But $\{b\}$ isn't given in our topology. I'm not understanding where the author got this singleton $\{b\}$ from. It's not defined in the topology so therefore it shouldn't be a problem, right?
He mentions $\{b\}$ because a space is $T_1$ iff all subsets of the form $\{x\}$ are closed. And indeed $\{b\}$ is not closed, so that is a counterexample: $X$ is not $T_1$, which is the point.