Hello;

I proved the statement that every submetrizable space has a regular

-diagonal. Kindly I want you to check if there is some mistake.

First a topological space

has a regular

-diagonal if the diagonal of

can be expressed as a countable intersection of closure of open sets

containing the diagonal, i.e. Δ

. A space

is called a submetrizable space if there is a continuous injective function from

into a metric space.

I wrote the proof in the file attached. Kindly open the file to check the proof and give me your comments. Every guidance or comment is highly appreciated.

Thaaaaaaaaank you in advance