Timeline for Are almost sequential spaces sequential?
Current License: CC BY-SA 4.0
27 events
| when toggle format | what | by | license | comment | |
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| Feb 6, 2019 at 10:33 | comment | added | Santi Spadaro | Ok, I've just added it. | |
| Feb 6, 2019 at 6:02 | comment | added | Dap | @SantiSpadaro: oh, wow. That should be an answer so it's more visible | |
| Feb 4, 2019 at 15:11 | comment | added | Santi Spadaro | $2^\kappa$ for uncountable $\kappa$ is a ZFC example of an almost sequential non-sequential space. To see why it's almost sequential, let $x$ be any point in $2^\kappa$ and let $D=\{y \in 2^\kappa: |\{\alpha < \kappa: x(\alpha) \neq y(\alpha) \}| < \aleph_0 \}$. Then $D$ is a sequential (even Fréchet-Urysohn) dense subspace of $2^\kappa$ which contains $x$. | |
| Feb 3, 2019 at 5:45 | history | edited | Dap | CC BY-SA 4.0 |
typo
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| Feb 3, 2019 at 5:21 | comment | added | Dap | @TarasBanakh: I think I've weakened the assumption to $\mathfrak{d}=\aleph_1.$ | |
| Feb 3, 2019 at 5:17 | history | edited | Dap | CC BY-SA 4.0 |
added argument with weaker assumption d=aleph_1
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| Feb 2, 2019 at 14:11 | history | edited | Dap | CC BY-SA 4.0 |
the example construction of f_\alpha made no sense
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| Feb 2, 2019 at 9:28 | vote | accept | Taras Banakh | ||
| Feb 2, 2019 at 9:28 | comment | added | Taras Banakh | @Dab Thank you for the corrections. Now it indeed works! Now it would be nice to weaken CH to something like $\mathfrak u\le\mathfrak d$. | |
| Feb 2, 2019 at 8:32 | comment | added | Dap | That should be "neighborhood subbase" in the last comment. Also I did spot one $\subseteq^*$ reversed in the "Specifically,.." sentence, fixed now. | |
| Feb 2, 2019 at 8:10 | history | edited | Dap | CC BY-SA 4.0 |
the subset was the wrong way round when referring to the construction in Jech
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| Feb 2, 2019 at 7:43 | history | edited | Dap | CC BY-SA 4.0 |
fix to previous edit
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| Feb 2, 2019 at 7:12 | comment | added | Dap | @TarasBanakh: Thanks for the comments! The sets $U_{n,\alpha}$ are not increasing in $\alpha$ - they just form a subbase of $*.$ The sequence $S_\alpha$ is $\supseteq^*$-increasing i.e. decreasing modulo finite sets. I've fixed the ordinal notation. | |
| Feb 2, 2019 at 7:01 | history | edited | Dap | CC BY-SA 4.0 |
fix successor ordinal notation + a slightly too strong assumption in the last point
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| Feb 1, 2019 at 10:22 | comment | added | Taras Banakh | And the main problem is with your definition of the topology at $*$: as $\alpha$ icreases so do the sets $U_{n,\alpha}$ but it should be vice-versa. If you replace $\le$ by $\ge$ in the definition of $U_{n,\alpha}$, then you will obtain the standard compact metrizable topology on $(\omega\times\{\omega\})\cup\{*\}$ and this will not allow you to prove that your space is not sequential. | |
| Feb 1, 2019 at 10:17 | comment | added | Taras Banakh | Thank you for another answer. I have some remarks: (1) it seems that the sequences $(S_\alpha)$ and $(f_\alpha)$ can be constructed under the assumtion $\mathfrak p=\mathfrak c$ (which is weaker than CH and follows from MA, actually it is equivalently to some version of MA); (2) usually by $\kappa^+$ the successor cardinal to $\kappa$ is denoted. So, your $\omega^+$ is a bit misleading; your can write $\omega+1$ or $\bar\omega$ instead. | |
| Feb 1, 2019 at 7:16 | history | edited | Dap | CC BY-SA 4.0 |
added 35 characters in body
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| Feb 1, 2019 at 7:11 | comment | added | Dap | @TarasBanakh: Sorry, I hadn't thought that through. I've now come up with a different construction using CH. | |
| Feb 1, 2019 at 7:05 | history | undeleted | Dap | ||
| Feb 1, 2019 at 7:05 | history | edited | Dap | CC BY-SA 4.0 |
new argument
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| Jan 29, 2019 at 12:57 | history | deleted | Dap | via Vote | |
| Jan 29, 2019 at 12:23 | comment | added | Taras Banakh | Thank you for the answer, but your third condition contradicts the second one. In fact, the second is true but the third is not. | |
| Jan 29, 2019 at 12:05 | history | undeleted | Dap | ||
| Jan 29, 2019 at 12:05 | history | edited | Dap | CC BY-SA 4.0 |
added 74 characters in body
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| Jan 29, 2019 at 11:58 | history | edited | Dap | CC BY-SA 4.0 |
added 153 characters in body
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| Jan 29, 2019 at 11:44 | history | deleted | Dap | via Vote | |
| Jan 29, 2019 at 11:18 | history | answered | Dap | CC BY-SA 4.0 |