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)

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)
  (Would resolve by a discrete space with cardinality $> \mathfrak c$, etc.)
• 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 become the second pair of completely pi-base-indistinguishable)
• S116 (Infinite broom) and S119 (Nested angles in the real plane)
  (would distinguish by P89 (Fixed point property))

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