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Identical Property/Space Merging/Distinguishment #910
Description
Activity
FYI, we discussed Markov Menger and sigma-relatively compact in the past, and decided to keep them separate. Our reasoning was: they are very different phrased properties, which a priori have nothing to do with each other. It seemed more instructive to keep them separate, with theorems showing the equivalence.
Reacted by Felix Pernegger and yhx-12243It's worth comparing with P152 where both a direct "topologically countable" and game-theoretic "Markov Rothberger" are given as aliases for the same (equivalent) property.
Note also that some implications like P157=>P156 might involve set-theoretic shenanigans, as P156 means Player One lacks a winning strategy for a game, and P157 means Player Two has a winning strategy, and these games can be indetermined due to the Axiom of Choice.
#1049 provides counter-example for P6
$\implies$ P61- changed the title
[-]Identical Property Merging[/-][+]Identical Property/Space Merging/Distinguishment[/+]on Dec 30, 2024 Note that S23 (Arens-Fort space) and S96 (Appert space) are not homeomorphic (which is not easy to see):
https://math.stackexchange.com/questions/5124885/are-the-appert-space-and-arens-fort-space-homeomorphicReacted by Patrick Rabau and yhx-12243
Similar for merging of homeomorphic spaces, I suppose to suggest merge two identical properties (i.e., both$A \implies B$ and $B \implies A$ can be deduced), because pi-base have aliases for properties and we can write equivalent definition in descripton.
For example, Completely regular and Uniformizable are in P12 now due to 52bcd9d.
Here is a list of identical properties which can be deduced by pi-base:
Merge P70 (Markov Menger) and P71 (sigma-relatively compact) #1684
Here is a list of$A \implies B$ but there are no counterexample for $B \implies A$ :
Trait Suggestion: Fort Space on the Real Numbers S154 is not Cozero complemented P61 #1049, Fort space on real numbers is not cozero complemented #1072
Theorem Suggestion: Connected+ strongly paracompact => countable extent #1189
Bing's example G does not have $G_\delta$ points and Michael's closed subspace is submetrizable #1375, (Resolved by S137 (Michael's subspace of Bing's Example G))
Similarly, for spaces, here is a list of pair of pi-base-indistinguishable spaces:
(would distinguish by P200 (Simply connected))
Footnotes
Means all traits of these two spaces (except that cannot decide in ZFC, etc.) are completed and they are same to both two spaces. ↩