On Buzyakovaʹs example; quotients of the Cantor trees

Let T be the Cantor tree and let A be a subset of the ωth level of T (= Cantor set C). Buzyakova considered the quotient space T A T ′ obtained from T × 2 by identifying two points 〈 a , 0 〉 and 〈 a , 1 〉 for each a ∈ A to construct an example of a non-submetrizable space of countable extent with a G δ -diagonal. We prove that the space T A T ′ is submetrizable if and only if C ∖ A is an F σ -set in C with the Euclidean topology. This improves Buzyakovaʹs Lemma.