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Tag Archives: uncountable
Homomorphisms from the harmonic archipelago group to finite groups
This post is a brief application of a result discussed in the last post about the existence of odd ways to map the fundamental group of the Hawaiian earring onto an arbitrary finite group : Theorem 1: Let be any nontrivial finite group and be a loop … Continue reading
Homomorphisms from the earring group to finite groups
One of the the surprising things about the earring group (the fundamental group of the earring space ) is that the group of homomorphisms to the additive group of integers is countable (see this post for details) even though is … Continue reading
Posted in earring group, earring space, Finite groups, Fundamental group, Group homomorphisms, Ultrafilter
Tagged anomalous, axiom of choice, cardinality, epimorphism, filter, finite group, fundamental group, Hawaiian earring, homomorphism, integers, nonprincipal ultrafilter, StoneCech compactification, uncountable
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The Uncountability of the Harmonic Archipelago Group
In a previous post, we discussed the fundamental group of the harmonic archipelago in some detail. One item that I skipped earlier is the fact that this group is uncountable. In this post, we’ll see why is uncountable. We’ll use … Continue reading