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Wikipedia states the following theorem:

Theorem (Mazurkiewicz): Let $(\mathcal{X},\rho)$ be a complete metric space and $A\subset\mathcal{X}$. Then the following are equivalent:

  1. $A$ is a $G_\delta$ subset of $\mathcal{X}$;
  2. There is a metric $\sigma$ on $A$ which is equivalent to $\rho|A$ such that $(A,\sigma)$ is a complete metric space.

I have tried Googling for proof of Mazurkiewicz theorem, but the result was I found two (seemingly) different theorems:

  1. The Hahn-Mazurkiewicz theorem, about which I found a Math SX question, a Wikipedia article and a pdf from a random website, plus a whole lot of other links;
  2. The Knaster-Kuratowski-Mazurkiewicz (or in short KKM) theorem, about which I found a pdf with proof and two previews, [1] and [2], which may well be the same pdf on two different links, but out of both of which the proof has been cut.

Now the first theorem is stated at the Math SX question and is totally different. It seems to be related to "space-filling curves", which I know nothing at all about. The other theorem, from what I read, is also rather different, since it has to do with a map's fixed points. So here I am. Can you post a proof of the Mazurkiewicz theorem above or a reference to one? Thx.

PS Feeling like only looking at the first page of Google results might not be judged enough research, I went on to page three, and found nothing about my theorem, seemingly. I did find what seems to be yet another couple of Mazurkiewicz theorems, none of which are mine. Then I got tired, thought I had searched well enough, posted the question and went to bed :).

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MickG
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1 Answers1

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It is a theorem of Alexandroff that if $X$ is a complete metric space and $A \subset X$ is a nonempty $G_\delta$ set, then $A$ can be remetrized with a complete metric.

The converse result is due to Mazurkiewicz: if $X$ is a metric space, $A \subset X$, and $A$ is homeomorphic to a complete metric space, then $A$ is a $G_\delta$ subset of $X$.

These are Theorems 10.18 and 10.19, respectively, in Real Analysis by Bruckner, Bruckner, and Thompson. Luckily is it available for free!

Umberto P.
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