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We recall that a topological space $(X,\tau)$ is submetrizable, if there is a coarser metrizable topology $\tau'$ that $\tau\supseteq\tau'$.

one of the properties of this these topological spaces is that each point of them is a $G_\delta$-point.

there are a lot of interesting topological spaces which are not metrizable.For example the surgenfery line denoted by $(\mathbb{R}_l)$ is a classical example of a normal lindelof nonmetrizable space in which every point of it is a $G\delta$-point. notice that this space is submetrizable.

another classical example is the moore plane denoted by $(\Gamma)$ which is non normal,non lindelof, separable space in which every point of it is a $G_\delta$-point.notice that this space is also a submetrizable space.

With the above summary, I can pose my Question.

Q. Is there an example of a topological space $(X,\tau)$ such that every point of it, is a $G_\delta$, but this space is not submetrizable?

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# Existence of a non-submetrizable topological space $(X, \tau)$

We recall that a topological space $(X,\tau)$ is submetrizable, if there is a coarser metrizable topology $\tau'$ that $\tau\supseteq\tau'$.

one of the properties of this topological spaces is that each point of them is a $G_\delta$-point.

there are a lot of interesting topological spaces which are not metrizable.For example the surgenfery line denoted by $(\mathbb{R}_l)$ is a classical example of a normal lindelof nonmetrizable space in which every point of it is a $G\delta$-point. notice that this space is submetrizable.

another classical example is the moore plane denoted by $(\Gamma)$ which is non normal,non lindelof, separable space in which every point of it is a $G_\delta$-point.notice that this space is also a submetrizable space.

With the above summary, I can pose my Question.

Q. Is there an example of a topological space $(X,\tau)$ such that every point of it, is a $G_\delta$, but this space is not submetrizable?