# Is each completely minimal topological group minimal?

A topological group $G$ is called

$\bullet$ minimal if it admits no strictly weaker Hausdorff group topology;

$\bullet$ completely minimal if it is Raikov-complete in each weaker Hausdorff group topology.

It is easy to see that each Raikov-complete minimal topological group is completely minimal and each completely minimal topological group is Raikov-complete.

Problem. Is each completely minimal topological group minimal? Equivalently, is an topological group completely minimal if and only if it is minimal and Raikov-complete?

Remark. The answer to this problem is affirmative for $\omega$-narrow topological groups of countable pseudocharacter and also for Abelian topological groups.

• check thm 3.7 in Categorically compact topological groups by Dikranjan and Uspenskij sciencedirect.com/science/article/pii/S0022404996001399 – Uri Bader May 31 '17 at 20:24
• @UriBader Thank you for the reference to the paper of Dikranjan-Uspenski. This theorem is the proof of the statement in Remark 2. So Problems 1,2 concerns injectivity of the homomorphisms. – Taras Banakh May 31 '17 at 22:02
• @ThomasRot The group $\mathbb Z$ dmits an injective homomorphism with non-closed image into the circle group. – Taras Banakh May 31 '17 at 22:04
• The definition is about $G$ which doesn't appear in the actual definition. – Asaf Karagila Jun 1 '17 at 10:57
• @Asasf-Karagila Thank you for the remark. I have corrected this misprint. – Taras Banakh Jun 1 '17 at 12:30