> I suggest starting with *Grundzuege der Mengenlehre* by F Hausdorff, and
> *Topologie* by P Aleksandrov and H Hopf, and the references in those. I
> think there's a translation into English of the book by Aleksandrov and
> Hopf, but I don't think the Hausdorff book has been translated.
The 1937 Third German edition of Felix Hausdorff's "Mengenlehre" was
translated into English by John R. Aumann et al and published by Chelsea
in 1962. Improbably a copy is in the engineering library at my work.
There is also a mention of a reprint by Chelsea in 1955 of the 1914 First
German edition of "Grundzu"ge der Mengenlehre", but no indication whether
this was a translation into English or not.
If you want, I can photocopy the "Further References" section, which gives
"an account of the origins of the more important concepts and theorems of
the subject".
Briefly, "open set", "neighborhood", and "complete space" occur in sections
22 and 23:
Section 22: open set: Lebesgue, Ann. Mat. Pura. Appl. (3) Vol 7 (1902)
page 242; Carathe/odory _Vorlesungen u"ber Reele Funktionen_ page 40.
In the first edition the open sets were called domains (Gebiete), a term
now used to denote connected open sets.
Section 23: Most of the concepts of this section derive from Cantor.
No, the index does not have an entry for "Hausdorff space".
I must quote the footnote on page 121:
The Lebesgue theory of measure and integration is not discussed
in this book, and acquaintance with it is not assumed. The reader
who does not understand the above allusion to the Riesz-Fischer
Theorem should simply forget it was mentioned.
James A Landau
Addendum:
Hausdorff says (p. 342) "the fundamental concepts of the theory of point
sets derive almost exclusively from [Cantor]."