TITLE: For a topological group $G$ and a subgroup $H$, is it true that $[\overline{H}, \overline{H}] = \overline{[H,H]}$? What about algebraic groups?
QUESTION [7 upvotes]: When discussing with awllower about this question, I begin to think about another one:

For a topological group $G$ and a subgroup $H$, is it true that $[\overline{H}, \overline{H}] = \overline{[H,H]}$? 

where $[H,H]$ denote the derived subgroups of $H$.
I think, if I define the map $\phi: G \times G \rightarrow G, (x,y) \mapsto xyx^{-1}y^{-1}$, then the question will become the equality between $\phi(\overline{H}\times \overline{H})$ and $\overline{\phi(H \times H)}$. Giving $G \times G$ the product topology, we will have $\phi(\overline{H} \times \overline{H}) \supseteq \overline{\phi(H \times H)}$ if $\phi$ is closed; and $\phi(\overline{H} \times \overline{H}) \subseteq \overline{\phi(H \times H)}$ because $\phi$ is continuous. (I hope I am not mistaken in thinking that $\overline{H \times H} = \overline{H} \times \overline{H}$, which is a condition of the above.) 
So, a derived question is:

For a topological group $G$, is the map $\phi: G \times G \rightarrow G, (x,y) \mapsto xyx^{-1}y^{-1}$ closed if $G \times G$ is given the product topology?


For the case when $G$ is an algebraic group, the topology will be changed to Zariski topology. Then is $\phi$ a closed map? Does $\phi(\overline{H}\times \overline{H})$ equal $\overline{\phi(H \times H)}$?
In fact I am more concerned with the algebraic group case. But if the first case is dealt, hopefully the algebraic case will become easier.
Thanks to everyone.

REPLY [6 votes]: For algebraic groups, $\overline{[H, H]}=[\overline{H}, \overline{H}]$. 
First $H\times H$ is dense in $\overline{H}\times \overline{H}$ for the Zariski topology (this is trivial for the product topology, but needs some proof for Zariski topology). So $[H, H]$ is dense in [$\overline{H}, \overline{H}]$ and $[\overline{H}, \overline{H}]\subseteq \overline{[H, H]}$. 
Second $[\overline{H}, \overline{H}]$ is closed. This is a general fact for algebraic groups: the derived group of an algebraic group is an algebraic group.