Identical Property/Space Merging/Distinguishment

[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:

• P70 (Markov Menger) and P71 (σ-relatively-compact)
  Merge P70 (Markov Menger) and P71 (sigma-relatively compact) #1684

Here is a list of $A \implies B$ but there are no counterexample for $B \implies A$:

• P61 (Cozero complemented) ⇒ P6 (T3.5)
  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
• P145 (Strongly paracompact) ⇒ P30 (Paracompact)
  Theorem Suggestion: Connected+ strongly paracompact => countable extent #1189
• P153 ($\omega$-Menger) ⇒ P66 (Menger)
• P70 (Markov Menger) ⇒ P72 (2-Markov Menger) ⇒ P69 (Strategic Menger)
• P91 (Eberlein compact) ⇒ P77 (Corson compact)
• P111 (Hemicompact) ⇒ P158 (Markov $k$-Rothberger) ⇒ P161 (Markov $k$-Menger)
• P166 (Has a coarser separable metrizable topology) ⇒ P112 (Submetrizable)
  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))
• P157 (Strategically $k$-Rothberger) ⇒ {P156 ($k$-Rothberger), P160 (Strategically $k$-Menger)} ⇒ P159 ($k$-Menger)

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Similarly, for spaces, here is a list of pair of pi-base-indistinguishable spaces:

• S23 (Arens–Fort Space) and S96 (Appert space) (completely pi-base-indistinguishable¹)
• S66 (Double origin plane) and S73 (Simplified arens square)
  (would distinguish by P200 (Simply connected))
• S116 (Infinite broom) and S119 (Nested angles in the real plane) (completely pi-base-indistinguishable)

Footnotes

1. Means all traits of these two spaces (except that cannot decide in ZFC, etc.) are completed and they are same to both two spaces. ↩

[prabau]

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.

[StevenClontz]

It'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.

[Moniker1998]

P6 (T3.5) $\iff$ P61 (Cozero complemented) is not true since if $D$ is uncountable and discrete, then $\beta D$ is cozero complemented but $\beta D\times \beta D$ is not
I'll try to see if there exist examples already on pi-base, though

[Moniker1998]

#1049 provides counter-example for P6 $\implies$ P61

[felixpernegger]

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-homeomorphic