Condition (7) of Theorem 4.54 in the book ``Axiom of Choice" by Horst Herrlich is the sentence: Each second countable topological space is separable.
Theorem 4.54 of this book says that (7) is equivalent to CC(R) where CC(R) is the axiom of countable choice for the real line R. I am sure that this is not true in a model for ZF because I can prove that (7) is equivalent to CC where CC is the axiom of countable choice. I believe that I can prove that CC(R) is equivalent in ZF to the statement that every second countable T_0-space is separable. I would be grateful if you could tell me whether someone else has ever noticed this mistake of H. Herrlich and published it somewhere. If the answer is YES, please, tell me where and by whom the correction of this Theorem 4.54 was published.
Dear Eliza,
Regarding your question about Horst's mistake I believe that it was just a typo. The reason I believe this is that I have proved this result and I have communicated it to him. See the the reference ker 200? in the references in his book. Unfortunately the status of the paper is still unknown. Recently Paul Howard ask for copy of this paper and I have prepapred a pdf file which I will forward to you via e-mail. I think that Horst told me once that some guy from Brazil also notice this typo. However, I do not know whether he has publish anything.
I thank you for your answer very much, as well as for your paper. I am grateful.
Hi, I guess I am the guy from Brazil. I guess that this little confusion is due to the following: CC is equivalent to "Every topological second-countable space is separable" and CC(R) is equivalent to "Every T_0 topological second-countable space is separable".
Dear Prof. Samuel Gomes da Silva,
Thank you for your answer very much. It makes my knowledge about Thm. 4.54 from the wonderful book by Horst Herrlich more complete. It is important for the history of mathematics.
Regards, Eliza Wajch
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