If $G$ is a paracompact topological group, then is $G \times G$ paracompact?


This question is raised by [Gepner and Henriques](https://arxiv.org/abs/math/0701916) (first paragraph of 2.2). Of course, this is not true for arbitrary paracompact spaces, as shown by the [Sorgenfrey plane](https://en.wikipedia.org/wiki/Paracompact_space#Examples).


Actually -- what's an example of a non-paracompact group?



If $G$ is a paracompact topological group, then is $G \times G$ paracompact?
I am at a [topology conference](https://sites.google.com/site/summertopology2018/) today, and among the many good topologists here is Jan van Mill, a leading expert on topological groups. I ran your question by him, thinking he might know the answer off the top of his head. He did -- the answer is that if $G$ is a paracompact topological group, then $G \times G$ need not be paracompact.


The construction of such a group can be found in



> 
> Yinhe Peng and Liuzhen Wu, "A Lindelöf group with non-normal square" ([link](https://www.researchgate.net/publication/321360206_A_LINDELOF_GROUP_WITH_NON-LINDELOF_SQUARE)) 
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> 
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Their main theorem is that there is a Hausdorff group $G$ such that $G$ is Lindelöf but $G \times G$ is not. Every Hausdorff group is regular, every regular Lindelöf space is paracompact, and every Hausdorff paracompact space is normal. Thus their group $G$ is paracompact, while $G \times G$ is not.


More information on Peng and Wu's work can be found in [these slides](https://www.fields.utoronto.ca/programs/scientific/14-15/forcing/slides/wednesday/Peng.pdf) of Yinhe Peng's.


