Is There a maximal space that is a P-space? we guess there is no maximal space which is also a P-space. Am I right? Do u know a counter example?
clarifications:
Maximal space is that space with topology $\tau$ which is maximal crowded topology on X.
crowded: a topology with no isolated point = dense in itself.
P-space:every $G_\delta$ set is open = every prime ideal is a Z-ideal = every prime ideal is a maximal ideal. (for others ref 4M of Gillman , Jerison)   
 A: Given a space $(X,\tau)$, the collection of all $G_\delta$-subsets of $X$ form a base for a stronger topology $\tau_\omega$ on $X$. It is easy to see that $(X,\tau_\omega)$ is a $P$-space (sometimes called the $G_\delta$ modification of $X$). If the original space $(X, \tau)$ is maximal, then there are two possibilities:
1) $(X, \tau_\omega)$ has isolated points. This will happen if and only if the original space has a point of countable pseudocharacter (i.e. a point which is a $G_\delta$).
2) $(X, \tau_\omega)$ is crowded. Then by maximality $\tau_\omega= \tau$ and therefore the original space is already a $P$-space.
It is known that if there are no measurable cardinals then every maximal space has countable pseudocharacter, so it cannot be a $P$-space.
I think it is still open whether there can be a [homogeneous] maximal space with uncountable pseudocharacter (see Question 168 [169] in Open Problems in Topology II). Note that by the remarks above, a maximal space in which every point has uncountable pseudocharacter must be a $P$-space.
