A space X is contractible onto ametric space provided there exists a one-to-one and continuous map from X onto a
metric space. A variety of diagonal conditions are used to establish necessary and
sufficient conditions in order that a space be contractible onto a metric space, and
some metrization theorems are also obtained, e.g.: (i) a separable space is
contractible onto a metric space if and only if has a zero-set diagonal; (ii) a
space X is metrizable if and only if X has a compatible semi-metric which is
uniformly continuous with respect to some product uniformity; (iii) a space X is
contractible onto a metric space if and only if X has a sequence G1,G2,⋯ of open
coverings which satisfies the following two conditions: (a) if x≠y, then there
exists n such that y∉ St (x,Gn); (b) if A,B ∈ Gn+1 and A ∩ B≠∅, then
there exists C ∈ Gn with A ∪ B ⊂ C; (iv) if Y is a perfectly normal space
which is the pseudo-compact image of a space which is contractible onto
a separable metric space, then Y is contractible onto a separable metric
space.
