Timeline for Question about closed projection
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| S Mar 16 at 18:12 | history | suggested | The Amplitwist | CC BY-SA 4.0 |
fixed broken link to Wikipedia
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| Mar 16 at 15:38 | review | Suggested edits | |||
| S Mar 16 at 18:12 | |||||
| Jul 15, 2010 at 9:50 | comment | added | BS. | And I overlooked the "second countable" hypothesis, which ruins my counterexamples. | |
| Jul 13, 2010 at 18:31 | comment | added | BS. | This is the following lemma : If projection $Y\times Z_1\to Z_1$ is closed and $Z_2$ is a subspace of $Z_1$, then $Y\times Z_2\to Z_2$ is also closed. Proof: take $F$ closed in $Y\times Z_2$ the projection $G$ of its closure $\overline{F}$ in $Y\times Z_1$ is closed in $Z_1$, so $G\cap Z_2$ is closed in $Z_2$. But since $\overline{F}\cap(Y\times Z_2)=F$ (opens of subspace are traces of opens of ambient space), this is the projection of $F$. | |
| Jul 13, 2010 at 16:38 | comment | added | Henno Brandsma | I thought about the latter too, as all countably compact space satisfy the fact that the projection onto R is closed (we can replace R by any sequential space even). But of course I realized we cannot have a second countable (note that condition! the OP works in second countable spaces) counterexample, as a second countable space is Lindelöf, and a Lindelöf and countably compact is compact... So the standard examples do not work. I also don't see why we can replace R by [0,1]: closedness is not preserved by embedding... So please elaborate on that? | |
| Jul 13, 2010 at 15:26 | history | edited | BS. | CC BY-SA 2.5 |
added 1047 characters in body; added 3 characters in body
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| Jul 13, 2010 at 14:36 | history | answered | BS. | CC BY-SA 2.5 |