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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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    but in the first theorem the first form requires to be hausdorff and the second not2012-11-01
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    If you follow the link, you'll see that (the typical use of) $T_3$ actually *implies* Hausdorff, so it's just redundant.2012-11-01
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    and wath occur with σ-locally discrete base and σ-discrete base2012-11-01
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    Likely just different sources using different terms for the same thing. You should check how the sources define the terms to make sure.2012-11-01
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    To see a proof of the equivalence of the Nagata-Smirnov and Bing characterizations of metrizability, look at Chapter 4, Theorem 18 of Kelley's _General Topology_.2012-11-01
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    @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.2013-04-12
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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