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The Structure of Locally Compact Connected Groups and Metrizability a
Author(s) -
ITZKOWITZ GERALD,
WU TA SUN
Publication year - 1993
Publication title -
annals of the new york academy of sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.712
H-Index - 248
eISSN - 1749-6632
pISSN - 0077-8923
DOI - 10.1111/j.1749-6632.1993.tb52520.x
Subject(s) - mathematics , metrization theorem , compact group , locally compact space , conjecture , element (criminal law) , group (periodic table) , pure mathematics , lie group , locally compact group , compact space , generalization , totally disconnected space , structured program theorem , duality (order theory) , simply connected space , discrete mathematics , mathematical analysis , separable space , chemistry , organic chemistry , political science , law
. A conjecture of Wilcox that states that a compact connected group is metrizable iff every element of the group is a metric element is considered. The stronger result that Wilcox's conjecture is valid for locally compact connected groups is proved. The method is to make diligent use of the structure theorem of Pontryagin‐Weil and the Iwasawa Decomposition Theorem. It turns out that the Pontryagin‐Weil structure theorem for compact connected groups yields a simple proof of the generalization of some well‐known facts for compact connected Lie groups. This in turn implies a quick proof of Mycielski's work with compact connected groups. Then making use of these facts, some structure theorems of Comfort, Robertson, and Soundararajan, and the duality theory for LCA groups, a number of theorems involving cardinal invariants of compact connected groups are established. These results imply the Wilcox theorem. Finally it is shown that the set of nonmetric elements in a compact connected group G is a dense nonmeasurable pseudocompact subset of G .