Timeline for Does ZF prove that proximity spaces are completely regular?
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5 events
| when toggle format | what | by | license | comment | |
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| Aug 30, 2011 at 19:13 | comment | added | Emil Jeřábek | Right, it is easy to verify from the definition that the formula above defines a proximity space inducing the original topology for any $T_4$ space. ZF proves that compact Hausdorff spaces are $T_4$. | |
| Aug 30, 2011 at 17:51 | comment | added | Emil Jeřábek | Every compact Hausdorff space is induced by a unique uniformity (whose entourages are simply all neighbourhoods of the diagonal); the proof of this does not need AC. I’m not really familiar with proximities, but I relied on Wikipedia’s word that, likewise, every compact Hausdorff topology is induced by a (unique) proximity space, namely $A\mathrel{\boldsymbol\delta}B$ iff $\overline A\cap\overline B\ne\varnothing$. Is there a problem with that? | |
| Aug 30, 2011 at 16:35 | comment | added | François G. Dorais | I've managed to convince myself that the order topology on $A$ is still given by a uniform structure. So I think this shows that uniformizability and complete regularity are not the same in this model. However, I'm not so enlightened when it comes to proximities (both specifically and in general). How does one define the proximity structure on $A$? | |
| Aug 29, 2011 at 19:35 | vote | accept | CommunityBot | ||
| Aug 29, 2011 at 19:25 | history | answered | Emil Jeřábek | CC BY-SA 3.0 |