pi-base
Novak space redefined
As mentioned before, Novak space have been ill-defined in Counterexamples due to possibility that $2^\mathfrak{c}$ is not a regular cardinal (as far as I know it's consistent with ZFC that it isn't), and didn't have a good definition on pi-base either.
The current definition is pulled directly from original article by Novak.
I didn't attempt to try to replicate the definition with a function $f:\beta\omega\to \beta\omega$ since I'm not sure if it even can be salvaged.
This PR attempts to fix that, together with replacing the properties stated by easy proofs.
There were 3 properties I replaced
- Not Lindelof was replaced by not compact (I can't imagine a proof which would not go through lack of compactness)
- Not meager was replaced by has isolated point
- Countably compact was replaced by weakly countably compact for better clarity (maybe)
Note that this is not an attempt to complete properties of this space, just to preserve them without much change so that they're clear.
@prabau you could review this if you want to familiarize yourself with the Novak space. Then perhaps do #1200 as well
Not sure I have time right at this moment, as I have never taken a look at it. But I'd like to get to it eventually.
Starting to look at this. It will take me a few days to have the time to read the paper, but here is some suggestion.
The README should keep the paragraph at the end referring to counterexample #112 in S&S, but mention why it is flawed and should be ignored.
@Moniker1998 I have reviewed this and it looks good overall. I just have a few minor formatting and other things, which I can mention after the conflicts get resolved. I imagine these could be due to the ref change that happened for S&S, changing from doi to zb.
Can you take care of that first? Maybe just need to merge the latest version of main?
@prabau I will need to figure out how to check those conflicts. On Github desktop I don't see any way to see them
I think it's due to some files being deleted, but were modified later on on the main branch now. Let me try to do this merge of the main branch into this branch.
The paper is pretty clear, I think, so I didn't look at S&S in detail for this. But where does it use that $2^\mathfrak c$ is a regular cardinal?
The paper is pretty clear, I think, so I didn't look at S&S in detail for this. But where does it use that 2^c is a regular cardinal?
I believe S&S uses that in point 1 of the properties, where it is supposed to construct it. So right at the beginning.
The proof in Novak constructs $x_\xi$ and $y_\xi$ by an alternating process which makes it not use regularity, I believe that S&S is trying to do it all at once which does use regularity.
Well you'd have to look in S&S