3 new alpha1 theorems#1781
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I want to emphasise, that we have had great progress on the alpha_i properties. When I joined in November, there was pretty much only First countable => alpha_1, but now we are getting very close to actually completing the properties. (Despite still having no nonterminal pi-base theorems about them!) :) |
prabau
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May 20, 2026
Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com>
prabau
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May 21, 2026
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| Utilising the metaproperties of {P210} and {P147}, we may assume that $X$ is {P57}. But then the assertion follows from {T238}, {T748}, {T894} and {T285}. |
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| Utilising the metaproperties of {P210} and {P147}, we may assume that $X$ is {P57}. But then the assertion follows from {T238}, {T748}, {T894} and {T285}. | |
| To prove the result it is enough to show every countable subspace of $X$ is {P210}. | |
| Note that each subspace of $X$ is also {P147}. | |
| So wlog we may assume $X$ is {P57}, and hence {P28} | |
| [(Explore)](https://topology.pi-base.org/spaces?q=Countable%2BP-space%2B%7EFirst+countable). | |
| And {T748}. |
prabau
reviewed
May 21, 2026
prabau
reviewed
May 21, 2026
Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com>
Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com>
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(I know I said I wouldnt open new PRs for now, but I really like the results here.)
This implements the missing theorems from this comment.
This gives two more traits, but very plausible many more (since for 14/17 remaining spaces either W-space, P-space or well based is still unknown).
We cant use explore to refer the the theorems used in the proof, otherwise the explore will just refer to the theorem itself, I tested this.
(Except for some reason for the third one it works as intended; to keep consistent and more stable to potential changes to pibase, lets keep it like it is however)
This PR has medium priority!