Cardinal numbers connected with approximate identities in Segal algebras (Q1824707)

The authors study the least cardinal number of a directed set defining an approximate identity in a Segal algebra on a locally compact Abelian group G. The main purpose of this note is to give an elementary proof of the well known result that G is metrizable if and only if the dual group \(\hat G\) is \(\sigma\)-compact (without using the duality theorem, the Plancherel theorem or the Cohen-Hewitt factorization theorem).