Separability of Topological Groups: A Survey with Open Problems
Round 1
Reviewer 1 Report
Minor corrections
page2
line 1 of Section 2, delete "of the form"
page 4
In corollary 5, G is B or vicecersa. Review the whole statement.
page 5
first line of Theorem 11 What does "countably productive" menas?
It does not sound right.
line -2 Revise "from Remark and Corollary"
line -1 Maybe delete "were" and write "is"
page 6
Corolary 14 The first C should be caligraphic. Also later in Remark 2.
page 11, In questions 5 through 7, are the conditions (iii) and (iV) independent?
Author Response
Minor corrections
page2 line 1 of Section 2, delete "of the form"
page 4 In corollary 5, G is B or vice versa. Review the whole statement.
page 5 first line of Theorem 11 What does "countably productive" means?
It does not sound right.
line -2 Revise "from Remark and Corollary"
line -1 Maybe delete "were" and write "is"
page 6 Corollary 14 The first C should be calligraphic. Also later in Remark 2.
Answer: DONE for all above
page 11, In questions 5 through 7, are the conditions (iii) and (iv) independent?
Answer: (iii) and (iv) are not independent as (iv) clearly implies (iii). A priori it is possible that there exist finite discrete, hence metrizable, groups which are not infinite.
Reviewer 2 Report
The article is an exhaustive and detailed survey of the topic of Separable Topological Groups. It summarises the results of several recent papers and lists a large number of open problems. It is an excellent reference for anybody interested in the subject and looking for directions of research.
I am fully happy for the article to be published in the present form, but I have a few suggestions, which the authors are free to take onboard or not. There are only two minor typos which should be corrected:
1) p11 l. 3 there should be a space between "groups. The"
2) p11 Problem 11, it should finish with a question mark.
Comments:
Many different notions are defined throughout the text, there are a couple I would personally like to see the definition reported:
p4. "uncountable cellularity"
p10 "barrelled"
p6 Should Remark 2 be stated as a Problem?
p7-8 From my understanding the results in 4.2 are obtained under some specific set theoretic conditions rather than just on the groups G and H. So I think the phrase in p7 line -15 (However by Theorem 22..." should be rephrased, or removed.
p9 Problem 4 (b) is the group (R,+) with the standard topology?
p14 Theorem 47, as that subrgoup G is mentioned a lot in the page, is it worth writing fully its construction/definition in the article?
Author Response
There are only two minor typos which should be corrected:
1) p11 l. 3 there should be a space between "groups. The"
2) p11 Problem 11, it should finish with a question mark.
Answer: DONE
Comments:
Many different notions are defined throughout the text, there are a couple I would personally like to see the definition reported:
p4. "uncountable cellularity"
p10 "barrelled"
Answer: DONE
p6 Should Remark 2 be stated as a Problem?
Answer: DONE
p7-8 From my understanding the results in 4.2 are obtained under some specific set theoretic conditions rather than just on the groups G and H. So I think the phrase in p7 line -15 (However by Theorem 22..." should be rephrased, or removed.
Answer: The correct sentence is: However by Theorem 23....
Groups G and H in Theorem 23 are constructed without any extra set theoretic assumptions.
p9 Problem 4 (b) is the group (R,+) with the standard topology?
Answer: DONE
p14 Theorem 47, as that subrgoup G is mentioned a lot in the page, is it worth writing fully its construction/definition in the article?
Answer: In fact, a construction of subgroup G of the product $T^{continuum}$ is quite difficult and involved.
It is not possible to describe this construction which goes by transfinite induction in a survey paper. We hope that the corresponding paper [5] with all technical details will be published soon.