Abstract
We prove the statement formulated in the title of the article. Then we apply it to show that there exists Lindelöf P-groups G and H satisfying w(G)=w(H)=|G|=|H|=ℵ1 such that G and H are not locally homeomorphic. This solves Problem 4.4.7 from the book (Arhangel'skii and Tkachenko, 2008 [1]) in the negative. Also, we present two homeomorphic complete Abelian P-groups one of which is ω-narrow and the other is not.
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