## General position properties satisfied by finite products of dendrites

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- by Philip L. Bowers
- Trans. Amer. Math. Soc.
**288**(1985), 739-753 - DOI: https://doi.org/10.1090/S0002-9947-1985-0776401-5
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## Abstract:

Let $\bar A$ be a dendrite whose endpoints are dense and let $A$ be the complement in $\bar A$ of a dense $\sigma$-compact collection of endpoints of $\bar A$. This paper investigates various general position properties that finite products of $\bar A$ and $A$ possess. In particular, it is shown that (i) if $X$ is an $L{C^n}$-space that satisfies the disjoint $n$-cells property, then $X \times \bar A$ satisfies the disjoint $(n + 1)$-cells property, (ii) ${\bar A^n} \times [ - 1,1]$ is a compact $(n + 1)$-dimensional ${\text {AR}}$ that satisfies the disjoint $n$-cells property, (iii) ${\bar A^{n + 1}}$ is a compact $(n + 1)$-dimensional ${\text {AR}}$ that satisfies the stronger general position property that maps of $n$-dimensional compacta into ${\bar A^{n + 1}}$ are approximable by both $Z$-maps and ${Z_n}$-embeddings, and (iv) ${A^{n + 1}}$ is a topologically complete $(n + 1)$-dimensional ${\text {AR}}$ that satisfies the discrete $n$-cells property and as such, maps from topologically complete separable $n$-dimensional spaces into ${A^{n + 1}}$ are strongly approximable by closed ${Z_n}$-embeddings.## References

- R. D. Anderson, D. W. Curtis, and J. van Mill,
*A fake topological Hilbert space*, Trans. Amer. Math. Soc.**272**(1982), no. 1, 311–321. MR**656491**, DOI 10.1090/S0002-9947-1982-0656491-8
P. L. Bowers, - Philip L. Bowers,
*Discrete cells properties in the boundary set setting*, Proc. Amer. Math. Soc.**93**(1985), no. 4, 735–740. MR**776212**, DOI 10.1090/S0002-9939-1985-0776212-6 - J. W. Cannon,
*Shrinking cell-like decompositions of manifolds. Codimension three*, Ann. of Math. (2)**110**(1979), no. 1, 83–112. MR**541330**, DOI 10.2307/1971245 - T. A. Chapman,
*Lectures on Hilbert cube manifolds*, Regional Conference Series in Mathematics, No. 28, American Mathematical Society, Providence, R.I., 1976. Expository lectures from the CBMS Regional Conference held at Guilford College, October 11-15, 1975. MR**0423357**
D. W. Curtis, - Robert J. Daverman,
*Detecting the disjoint disks property*, Pacific J. Math.**93**(1981), no. 2, 277–298. MR**623564** - Robert J. Daverman and John J. Walsh,
*Čech homology characterizations of infinite-dimensional manifolds*, Amer. J. Math.**103**(1981), no. 3, 411–435. MR**618319**, DOI 10.2307/2374099 - Tadeusz Dobrowolski and Henryk Toruńczyk,
*On metric linear spaces homeomorphic to $l_{2}$ and compact convex sets homeomorphic to $Q$*, Bull. Acad. Polon. Sci. Sér. Sci. Math.**27**(1979), no. 11-12, 883–887 (1981) (English, with Russian summary). MR**616181** - James Dugundji,
*Topology*, Allyn and Bacon, Inc., Boston, Mass., 1966. MR**0193606**
R. D. Edwards, - Witold Hurewicz and Henry Wallman,
*Dimension Theory*, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR**0006493** - Jan van Mill,
*A boundary set for the Hilbert cube containing no arcs*, Fund. Math.**118**(1983), no. 2, 93–102. MR**732657**, DOI 10.4064/fm-118-2-93-102 - Frank Quinn,
*Ends of maps. I*, Ann. of Math. (2)**110**(1979), no. 2, 275–331. MR**549490**, DOI 10.2307/1971262 - K. Sieklucki,
*A generalization of a theorem of K. Borsuk concerning the dimension of $ANR$-sets*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.**10**(1962), 433–436. MR**198430** - H. Toruńczyk,
*On $\textrm {CE}$-images of the Hilbert cube and characterization of $Q$-manifolds*, Fund. Math.**106**(1980), no. 1, 31–40. MR**585543**, DOI 10.4064/fm-106-1-31-40 - H. Toruńczyk,
*Characterizing Hilbert space topology*, Fund. Math.**111**(1981), no. 3, 247–262. MR**611763**, DOI 10.4064/fm-111-3-247-262 - Stephen Willard,
*General topology*, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR**0264581**

*Applications of general position properties of dendrites to Hilbert space topology*, Ph.D. Dissertation, Univ. of Tennessee, 1983.

*Boundary sets in the Hilbert cube*, preprint. —,

*Preliminary report, boundary sets in the Hilbert cube and applications to hyperspaces*, preprint.

*Approximating certain cell-like maps by homeomorphisms*, Abstract preprint. See also Notices Amer. Math. Soc.

**24**(1977), A649, #751-G5.

## Bibliographic Information

- © Copyright 1985 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**288**(1985), 739-753 - MSC: Primary 54F50; Secondary 54C25, 54C35, 54F35
- DOI: https://doi.org/10.1090/S0002-9947-1985-0776401-5
- MathSciNet review: 776401