I am studying metrization. I encountered different formulations of the theorems from different parts of my reading. For example, in the Nagata–Smirnov metrization theorem, I found:
A topological space $X$ is metrizable if and only if it is $T_3$ and Hausdorff and has a $\sigma$-locally finite base, and other: $X$ is $T_3$ , and there is a $\sigma$-locally finite base for $X$.
For the Bing metrization theorem, I found:
$X$ is $T_3$, and there is a $\sigma$-locally discrete base for $X$. And this other: a space is metrizable if and only if it is regular and $T_0$ and has a $\sigma$-discrete base.
Which is the true form?
All of them. They are equivalent. Part of the confusion may stem from the fact that the $T_n$ notation isn't always used the same way. See here for more detail on their interrelationships.