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Misra's subspace of E_0 is totally separated #1196

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merged 3 commits into from
Jan 14, 2025
Merged

Misra's subspace of E_0 is totally separated #1196

merged 3 commits into from
Jan 14, 2025

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Moniker1998
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@Moniker1998 Moniker1998 commented Jan 12, 2025

Misra's subspace of E_0 is not extremally disconnected.

Also see #1060

@Moniker1998 Moniker1998 changed the title Mistra's subspace of E_0 is not extremally disconnected Misra's subspace of E_0 is not extremally disconnected Jan 12, 2025
@prabau
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prabau commented Jan 13, 2025

An equivalent way, maybe slightly easier to present, is to give two disjoint open sets with intersecting closures.
In addition to $U$, let $V$ be the set of $a_{\alpha\beta}$ with $\alpha\in\omega_1\setminus A$. Then $a\in\overline{U}\cap\overline V$. But what you have is ok too.

Could you add the closely related trait P48 (totally separated) = true? (easy)

@prabau prabau self-requested a review January 13, 2025 05:01
@Moniker1998
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Moniker1998 commented Jan 13, 2025

@prabau it turns out that the theorem I want to add #1195 here already implies this space is not extremally disconnected

So instead, I will replace it by totally separated.

See also #1059

@prabau prabau changed the title Misra's subspace of E_0 is not extremally disconnected Misra's subspace of E_0 is totally separated Jan 14, 2025
@prabau prabau merged commit b33baf7 into main Jan 14, 2025
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@prabau prabau deleted the misra's-subspace branch January 14, 2025 07:50
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2 participants