pi-base
Adding cozero completemented (P61) to various spaces
I mainly used the argument from #1518 by @prabau. This completes S86, S84 and S24.
The proposition from #1518 is probably not exhausted yet (i.e. can be applied to more spaces), I just took out all the easy examples with this PR.
@felixpernegger I have added Proposition 1 in https://math.stackexchange.com/questions/5108293 to give details on the implication [ X cozero complemented => same for its Tychonoff reflection ]. Please double check that. So we can refer to it when we need the contrapositive.
I am making a corresponding suggestion for S24 below. More details, but it seems acceptable to me. Please let me know what you think.
(also changed the previous Proposition to Proposition 2)
@felixpernegger I have added Proposition 1 in https://math.stackexchange.com/questions/5108293 to give details on the implication [ X cozero complemented => same for its Tychonoff reflection ]. Please double check that. So we can refer to it when we need the contrapositive.
I am making a corresponding suggestion for S24 below. More details, but it seems acceptable to me. Please let me know what you think.
(also changed the previous Proposition to Proposition 2)
It looks good to me, just one thing I wanna ask to make sure: Given some continuous maps $X \to \mathbb{R}$, this always factors over $X_{Tych}$ right? (since we only have to check preimages of (countably many) rational intervals of $\mathbb{R}$ and these are cozero sets of $\mathbb{R}$ basically.
@felixpernegger I have added Proposition 1 in https://math.stackexchange.com/questions/5108293 to give details on the implication [ X cozero complemented => same for its Tychonoff reflection ]. Please double check that. So we can refer to it when we need the contrapositive. I am making a corresponding suggestion for S24 below. More details, but it seems acceptable to me. Please let me know what you think. (also changed the previous Proposition to Proposition 2)
It looks good to me, just one thing I wanna ask to make sure: Given some continuous maps X → R , this always factors over X T y c h right? (since we only have to check preimages of (countably many) rational intervals of R and these are cozero sets of R basically.
If this is true, it would give a nice universal property
Given some continuous maps $X \to \mathbb{R}$, this always factors over $X_{Tych}$ right?
That is by the universal property in the definition of a reflective subcategory. In this case, the Tychonoff spaces together with continuous functions form a reflective subcategory of the full category of topological spaces. Of course, this has to be proved, but it's not that bad, there are a few mathse posts about it I think.
Given that, all we need is that $\mathbb R$ is Tychonoff. So automatically $f:X\to\mathbb R$ factors (uniquely) through $\varphi:X\to X_{Tych}$.
I've added one line to the README of S24 for convenience in accessing the related space.
Btw, I am using gihub.dev for that and I noticed lately that I am having trouble editing or entering commit messages sometimes. The whole github editor freezes somehow. When I refresh the browser page, it starts working again. Have you seen something similar by any chance?
I've added one line to the README of S24 for convenience in accessing the related space.
Btw, I am using gihub.dev for that and I noticed lately that I am having trouble editing or entering commit messages sometimes. The whole github editor freezes somehow. When I refresh the browser page, it starts working again. Have you seen something similar by any chance?
I always use VSCode with Github Desktop now (which works very well), so im afraid I cant help you