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)
1 Answer
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.
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$\begingroup$ Dear Ramiro de la Vega. Can u address where the statement "if there are no measurable cardinals then every maximal space has countable pseudocharacter" mentioned. $\endgroup$ Sep 11, 2013 at 9:15
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$\begingroup$ In defining a P-space, Always we suppose the space X is tychonoff at first. If we edit question as follows: Is There a maximal tychonoff space that is a P-space? your answer dosen't work. because $(X,\tau_{\omega})$ need not be a tychonoff space. $\endgroup$ Sep 11, 2013 at 11:10
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$\begingroup$ @Vahideh: You can find the statement in the paragraph right before Question 168 in the reference I gave: If a maximal space has a point of uncountable pseudocharacter, then that pseudocharacter and the cardinality of the space are both a measurable cardinal. $\endgroup$ Sep 11, 2013 at 15:15
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3$\begingroup$ @Vahideh: Hewitt showed that a maximal Tychonoff space is extremally disconnected. Isbell showed that an extremally disconnected $P$-space of nonmeasurable cardinality is discrete. So the situation is pretty much the same for maximal Tychonoff spaces. $\endgroup$ Sep 11, 2013 at 15:39
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