Next Article in Journal
Lipschitz Stability for Non-Instantaneous Impulsive Caputo Fractional Differential Equations with State Dependent Delays
Next Article in Special Issue
Hereditary Coreflective Subcategories in Certain Categories of Abelian Semitopological Groups
Previous Article in Journal
Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV
Previous Article in Special Issue
(L)-Semigroup Sums
 
 
Article
Peer-Review Record

Separability of Topological Groups: A Survey with Open Problems

by Arkady G. Leiderman 1,*,† and Sidney A. Morris 2,3,†
Reviewer 1: Anonymous
Reviewer 2: Anonymous
Submission received: 25 November 2018 / Revised: 23 December 2018 / Accepted: 25 December 2018 / Published: 29 December 2018
(This article belongs to the Collection Topological Groups)

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.

 

 

 

Back to TopTop