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**student2011** Hello;

I proved the statement that every submetrizable space has a regular $\displaystyle G_\delta$-diagonal. Kindly I want you to check if there is some mistake.

First a topological space $\displaystyle X$ has a regular $\displaystyle G_\delta$-diagonal if the diagonal of $\displaystyle X$ can be expressed as a countable intersection of closure of open sets $\displaystyle U_{n}$ containing the diagonal, i.e. Δ$\displaystyle =\cap_n{cl({U_n})}$. A space $\displaystyle X$ is called a submetrizable space if there is a continuous injective function from $\displaystyle X$ 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