Timeline for are acyclic fibrations of nice spaces absolute extensors for perfectly normal spaces?
Current License: CC BY-SA 4.0
13 events
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| Nov 21, 2021 at 13:48 | comment | added | Tyrone | See the book Continuous Selections of Multivalued Mappings by Repovs and Semenov, and consult Th.5.22 on pg. 291. Given what I've told you in the previous paragraph, the only nontrivial thing to prove is that the family $\{f^{-1}(y)\}$ is ELC$^n$ for any $n$, and this is contained elsewhere in the book. | |
| Nov 21, 2021 at 13:15 | comment | added | user420620 | thanks! what is a reference for "Any such map between finite complexes is already soft for the class of (finite-dimensional) paracompacta." ? | |
| Nov 21, 2021 at 11:03 | comment | added | Tyrone | The case for finite complexes is probably already known. Every finite complex is a metrisable ANR, and each acyclic Serre fibration between such is a Hurewicz fibration and a homotopy equivalence. Any such map between finite complexes is already soft for the class of (finite-dimensional) paracompacta. The jump from here to perfectly normal spaces should be doable, but I'm afraid that I'm not the person to fill in the gaps. | |
| Nov 21, 2021 at 10:55 | comment | added | Tyrone | The paper Countable spaces without extension properties contains the construction of $X$. I couldn't find a copy online, but I'd be grateful if anyone does track down a digital copy. | |
| Nov 21, 2021 at 9:36 | comment | added | user420620 | Following up the names you gave, found a maybe useful result in (van Douwen, E. K.; Lutzer, D. J.; Przymusi\'nski, T. C. Some extensions of the Tietze-Urysohn theorem. Amer. Math. Monthly 84 (1977), no. 6, 435--441. (Reviewer: J. Dugundji) 54E40 (54E35)): for A closed in X normal, there is a continuous map $C^{bounded}(A)\to C^{bounded}(X)$ where $C^{bounded}(X)$ denotes the space of bounded continous functions on $X$.. Does this result belong a result to "selection theory" you refer to ? | |
| Nov 21, 2021 at 9:29 | comment | added | user420620 | Many thanks, interesting! Though I was not able to find online their paper, presumably van Douwen, Eric K.; Pol, Roman Countable spaces without extension properties. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), no. 10, 987--991. (Reviewer: D. J. Lutzer) 54C20. Frankly, I do find it rather surprising if the statement isn't known (either proof or some standard counterexample, for finite complexes...) | |
| Nov 20, 2021 at 18:01 | comment | added | Tyrone | On the other hand, if you restrict to (suitably nice) acyclic Serre fibrations between (suitably nice) finite complexes, then something should be true. How far you get with this will probably depend on your knowledge of selection theory. | |
| Nov 20, 2021 at 17:55 | comment | added | Tyrone | Let $C\mathbb{N}$ be the cone over a countably infinite discrete complex (this is a contractible 1-dimensional polyhedron). van Douwen and Pol have constructed a countable regular $T_2$ space $X$ (which is thus perfectly normal) and a function $A\rightarrow C\mathbb{N}$, defined on a certain closed $A\subseteq X$, which does not extend over any neighboourhood in $X$. In particular, the map of countable complexes $C\mathbb{N}\rightarrow\ast$ is both a Hurewicz fibration and a homotopy equivalence, but is not soft wrt all perfectly normal pairs. | |
| Nov 19, 2021 at 18:58 | history | edited | user420620 | CC BY-SA 4.0 |
added references to soft maps by E.Scshepin
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| Nov 19, 2021 at 18:26 | comment | added | user420620 | There is a notion of soft map by Eugene Schepin , e.g. see E. V. Shchepin, Soft maps of manifolds, Uspekhi Mat. Nauk, 1984, Volume 39, Issue 5(239), 209–224, or A. Chigogidze, Inverse Spectra, North Holland, Amsterdam, 1996, and reference therein. I haven't yet understood whether they are related to this particular question. | |
| Nov 17, 2021 at 10:13 | history | edited | user420620 | CC BY-SA 4.0 |
rewrote using the terminogy "absolute extensor"
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| Nov 15, 2021 at 18:31 | history | edited | user420620 | CC BY-SA 4.0 |
added 8 characters in body
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| Nov 15, 2021 at 18:23 | history | asked | user420620 | CC BY-SA 4.0 |