Does every element of the derived series of a pro-p group is also a pro-p group?


The problem reduces to showing that every element of the derived series is a closed subgroup...But is it always true?


Hope you'll be able to help me


Thanks ! 



Derived Series of Pro-p groups
It seems this can't be expected. From [Simons' thesis](http://ora.ouls.ox.ac.uk/objects/uuid%253A01075c36-c7e6-4def-9647-86b4346e4726/datastreams/THESIS01) (p. xii): 



> 
> The derived group of any finitely generated profinite group is closed, but Roman'kov [29] has provided an example of a finitely generated pro-p group in which the second derived group is not closed. 
> 
> 
> 


I haven't Roman'kov's paper at hand yet and don't know about details of his example. The reference is: 


Roman'kov: The width of verbal subgroups of solvable groups. Algebra i Logika 21(1),60-72(1982). 


In contrast, the groups in the lower central series of a finitely generated profinite group are always closed. This is proved by Nikolov-Segal in [this paper](http://annals.math.princeton.edu/wp-content/uploads/annals-v165-n1-p05.pdf)  (Theorem 1.4). 


