Locally homeomorphic infinite Lindelof P-groups are homeomorphic

We prove the statement formulated in the title of the article. Then we apply it to show that there exists Lindel & ouml;f P-groups G and H satisfying w(G) = w(H) = |G| = |H| = aleph 1 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 omega-narrow and the other is not. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.