Cellularity and the index of narrowness in topological groups

It is well known that the cellularity, c(H), and the index of narrowness, in(H), of an arbitrary topological group H satisfy the inequalities in(H) ≤ c(H) and c(H) ≤ 2^in(H). Here in(H), the index of narrowness of H, denotes the minimal cardinal number κ ≥ ω such that the group H can be covered by at most κ translates of any neighborhood of the identity element in H.

We describe the relations between the cellularity and index of narrowness in topological groups and their Gδ-modifications.

Also, we present some bounds for the complexity of continuous real-valued functions f on an arbitrary ω-narrow group G, defined to be the least cardinal τ ≥ ω such that there exists a continuous homomorphism π : G → H onto a topological group H with w(H) ≤ τ such that π ≺ f.

It is shown that this complexity is not greater than 2^2^ω and, if G is weakly Lindel ̈of (or 2^ω-steady), then it does not exceed 2^ω

A special attention is paid to R-factorizable groups, that is, to the groups whose complexity is countable. Several open problems about R-factorizable group are discussed.