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I studying metrization and in different parts I encountered different formulations of theorems, 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?

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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.

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    @Henno: Right. I wasn't intending to say that (locally) finite and (locally) discrete were the same. I was just pointing out that the metrization theorems themselves were equivalent, and that the "different" versions of the same theorems likely sprung from different usages of the same term (like $T_3$), or use of different terms for the same thing ($\sigma$-locally discrete or $\sigma$-discrete).2013-04-12