# Thread: every submetrizable space has a regular G_delta diagonal

1. ## every submetrizable space has a regular G_delta diagonal

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.

2. ## Re: every submetrizable space has a regular G_delta diagonal

Originally Posted by 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.