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.
• If you follow the link, you'll see that (the typical use of) $T_3$ actually implies Hausdorff, so it's just redundant. Commented Nov 1, 2012 at 14:56
• @CameronBuie they're not the same thing, a locally discrete family means that every point in the space has a neighbourhood that intersects at most one (not just finitely many) of the members. Bing preferred to work with $\sigma$-locally discrete families, Nagata and Smirnov with $\sigma$-locally finite ones. Having one type of base implies having the other type and vice versa (for regular spaces) hence the equivalence. They're not necessarily the same base, though, it takes some work to get from locally finite to locally discrete. Commented Apr 12, 2013 at 9:07