Symbols w(X), nw(X), and hl(X) denote the weight, the network weight, and the hereditary Lindelf number of a space X, respectively.
We prove the following factorization theorems.
(1)
Let X and Y be Tychonoff spaces, ϕ: X→Y a continuous mapping, hl(X)≤τ, and w(Y)≤τ. Then there exist a Tychonoff space Z and
continuous mappings ψ: X→Z, χ: Z→Y such that ϕ=χ o ψ, Z=ψ(X), w(Z)≤τ andind Z≤ind X. Moreover, if nw(X)≤τ, then mapping ψ is one-to-one.
(2)
Let π: G→H be a continuous homomorphism of a Hausdorff topological group G to a Hausdorff topological group H, hl(G)≤τ and
w(H)≤τ. Then there are a Hausdorff topological group G* and continuous homomorphisms g: G→G*, h: G*→H so that π=h o g, G*=g(G), w(G*)≤τ andind G*≤ind G. If nw(G)≤τ, then g is one-to-one.
(3)
For every continuous mapping ϕ: X→Y of a regular Lindelf space X to a Tychonoff space Y one can find a Tychonoff space Z
and continuous mappings ψ: X→Z, χ: Z→Y such that ϕ=χ o ψ, Z=ψ(X), w(Z)≤w(Y),dim Z≤dim X, andind
0 Z≤ind
0 X, whereind
0 is the dimension function defined by V.V.Filippov with the help of Gδ-partitions. If we additionally suppose that X has a countable network, then ψ can be chosen to be one-to-one. The analogous
result also holds for topological groups.
(4)
For each continuous homomorphism π: G→H of a Hausdorff Lindelf Σ-group G (in particular, of a σ-compact group G) to a Hausdorff
group H there exist a Hausdorff group G* and continuous homomorphisms g: G→G*, h:G*→H so that π=h o g, G*=g(G), w(G*)≤w(H),dimG*≤dimG, andind G*≤ind G. Bibliography: 25 titles.