Continuous fuzzy groups (Q1920355)

A fuzzy group \((G,\mu)\) is said to be continuous if \(G\) is a topological group and \(\mu: G\to [0,1]\) is continuous. The author defines a topological group \(G\) to be fuzzy trivial if all continuous functions \(\mu\) from \(G\) to \([0,1]\) such that \(\mu\) is a fuzzy subgroup of \(G\) are constants. Some of the interesting results obtained by the author are: (i) A connected topological group is fuzzy trivial; (ii) a totally disconnected locally compact group is fuzzy nontrivial; (iii) if \((G,\mu)\) is a continuous fuzzy group with compact \(G\), then (a) \(\text{Im} (\mu)\) is either a finite set or a countable set with a single limit point \(\mu(e)\), and (b) if \(G\) is totally disconnected, then \(G\) is finite iff \(\text{Im} (\mu)\) is a finite set for any \(\mu\).