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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 not –  Fenrir Nov 1 '12 at 14:54
    
If you follow the link, you'll see that (the typical use of) $T_3$ actually implies Hausdorff, so it's just redundant. –  Cameron Buie Nov 1 '12 at 14:56
    
and wath occur with σ-locally discrete base and σ-discrete base –  Fenrir Nov 1 '12 at 15:02
    
Likely just different sources using different terms for the same thing. You should check how the sources define the terms to make sure. –  Cameron Buie Nov 1 '12 at 15:06
    
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. –  Scott L Nov 1 '12 at 16:09

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