My definition for $T_3$ space is a topological space such that 'For any point $x$ and a closed subset $A$, there exist nonoverlapping neighborhoods'.

(This is the definition of a regular space in "Topology - Munkres")

(p.215) Urysohn Metrization theorem: Every second countable $T_3$ space $X$ is metrizable.

Is it true? I can prove this with an additional condition, that is, $X$ is $T_0$. However, i cannot prove this without this condition. How do i prove this without this condition?

What are necessary conditions for the Urysohn Metrization theorem?

  • $\begingroup$ The indiscrete space $[0,1]$ is second-countable and regular, but clearly not metrizable. So you will need the Hausdorffness :-) $\endgroup$ – Stefan Hamcke Oct 22 '13 at 21:28
  • $\begingroup$ Check the definition of 'regular' on page 195 of the book. It requires the space to be $T_1$, which implies Hausdorff. $\endgroup$ – Ayman Hourieh Oct 22 '13 at 21:30
  • $\begingroup$ @Stefan Thank you! I think $T_0$ is essential, $T_2$(Hausdorff) is too large. $\endgroup$ – Jj- Oct 22 '13 at 21:32
  • $\begingroup$ @Jj-: For regular spaces $T_0=T_1=T_2$. $\endgroup$ – Stefan Hamcke Oct 22 '13 at 21:47

You can’t prove it without that condition. There are, unfortunately, two exactly opposite usages of the terms $T_3$ and regular. In my usage, which I think is the more common, the numerical $T_n$ properties form a genuine hierarchy: a $T_3$ space is automatically $T_2$, $T_1$, and $T_0$. Specifically, it’s a $T_1$ space in which points and closed sets can be separated by disjoint open sets. That’s the sense in which $T_3$ is used in the metrization theorem. What you call $T_3$ is what I would call simply regular, so that for me a $T_3$ space is a regular $T_1$ space. Unfortunately, as I said, some people reverse the terms, a practice that naturally engenders a certain amount of confusion. The discussion in Wikipedia reasonably clear on this point.

  • $\begingroup$ Thank you! But i prefer 'proofwiki' definitions to 'wikipedia' ones. $\endgroup$ – Jj- Oct 22 '13 at 21:35
  • $\begingroup$ @Jj-: I don’t. Wikipedia is generally quite good on mathematical topics, and I dislike the Pr$\infty$fWiki style (though to be fair, that applies more to the proofs, than to the definitions: in my view the former are written in a very opaque, unhelpful style). And in this case I consider the Pr$\infty$fWiki definition simply wrong: it represents a minority usage and a somewhat illogical one, since it destroys the hierarchical nature of the $T_n$ system. You’re welcome! $\endgroup$ – Brian M. Scott Oct 22 '13 at 21:45

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