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 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?



Existence of  a non-submetrizable topological space $(X, \tau)$
The Alexandroff´s double arrow space (i.e. $[0,1] \times \{0,1\}$ with the lexicographic order) is first countable (so any point is a $G\_\delta$), but is compact and non-metrizable (hence it is not submetrizable).


