May 22 and 23, 1930
Dear Dr. Whyburn:
I am glad to have your letter of April 28. I was glad, too, to receive those of earlier dates and interested to hear of the progress of your investigations.
You say that Knaster questioned you concerning my proof of the arcwise connectivity of connected and connected im kleinen Gd -sets. In December, 1926, I presented to the American Mathematical Society a communication of which the abstract is printed on page 141 of vol. 33 (1927) of The Bulletin of The American Mathematical Society. Thinking that possibly you may not have a copy of this number of the Bulletin at hand I will copy from this abstract:
"Axioms 1, 2 and 4 and Theorem 4 of the author's Foundations of plane analysis situs hold true in euclidean space of any finite number of dimensions. But Axiom 4 does not hold true in a Hilbert space or a space Ew , and Axiom 1 does not hold in the non-separable space Dw. Axiom 2, Theorem 4, and Axiom 1' stated below hold true in all these spaces, and form a basis for a considerable body of theorems, in particular, Theorems 1-10, and 15, of the paper cited above. Axiom 1' : There exists a countable sequence G1, G2, G3,... such that (a) for each n, Gn is a collection of regions covering space, (b) if P1 and P2 are distinct points of a region R, there exists an integer d such that if n > d and Kn is a region containing P1 and belonging to Gn, then Kn' is a subset of R-P2, (c) if R1, R2, R3,... is a sequence of regions such that, for each n, Rn belongs to Gn and such that, for each n, R1, R2,..., Rn have a point in common, then there exists a point common to all the point sets R1' , R2' , R3' ,...."
This is one of a number of theorems which I have stated, beginning, I believe, at about that time, but whose proofs I have refrained from publishing, preferring to publish them for the first time in my Colloquium book. Theorem 15 of my paper On the foundations of plane analysis situs states that if A and B are distinct points of a domain M, there exists a simple continuous arc from A to B that lies wholly in M. Thus, in the above quoted abstract, I stated, in effect, among other things, that, in every space satisfying Axioms 1' and 2 and Theorem 4 of F. A., every connected domain is arcwise connected. But it is easy to see that if M is any connected and connected im kleinen Gd set lying in a space satisfying these axioms (and Theorem 4) then M, regarded as a space, itself satisfies these axioms (and Theorem 4 unless it is a single point). Hence every connected and connected im kleinen Gd -set lying in such a space is arcwise connected and arcwise connected im kleinen. I can show that there exists spaces satisfying Axioms 1' and 2 and Theorem 4 of F. A. but which are not metric. Thus my theorem is more general than the theorem that in every complete metric space every connected and connected im kleinen Gd -set is arcwise connected and arcwise connected im kleinen. I will enclose herewith a proof of the theorem, stated in my Bulletin abstract of 1927, to the effect that, in every space satisfying Axioms 1' and 2 and Theorem 4 of F. A., every connected domain is arcwise connected. This proposition remains true if Axiom 1' is replaced by any one of a number of other modifications of Axiom 1 which I have formulated, including those mentioned in my Boulder lectures. Will you please give the enclosed sheets to Doctor Knaster and also, if you wish, show him this letter?
Work on my book is progressing slowly but I hope to send off the manuscript before Jan. 1, 1931. I have been delayed considerably since about March 15 by various matters including (1) the preparation and delivery of five University Research Professor Lectures and (2) Cleveland's and Dorroh's theses. Dorroh passed his oral examination Monday of this week and Cleveland passed his the next day. They are scheduled to receive their degrees in June. Roberts is to [be] back here next session. Vickery and Klipple are doing quite well. I have considerable hopes for both of them.
Kindly remember me to Mrs. Whyburn and to Drs. Knaster, Kuratowski, Mazurkiewicz, Sierpinski and Zanrankiewicz. I am glad you and Mrs. Whyburn are planning to come to Texas in September. I am looking forward to seeing you.