$Z_n$-sets and the disjoint $n$-cells property in products of ANR's

T.O. Banakh; Kh.R. Trushchak

It is shown that for $X,Y\in$ANR the product $A\times B$ of a $Z_n$-set $A\subset X$ and a $Z_m$-set $B\subset Y$ is a $Z_{n+m+1}$-set in $X\times Y$. This implies that the product $X\times Y$ of a~$\sigma Z_n$-space $X\in ANR$ and a~$\sigma Z_m$-space $Y\in ANR$ is a $\sigma Z_{n+m+1}$-space. Also we show that for every non-degenerate absolute retract $X$ and every $n$

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3. K. Borsuk, Theory of retracts, PWN, Warsaw, 1967.

4. T. Dobrowolski, The compact $Z$-set property in convex sets, Top. Appl. 23 (1986), 163--172.

5. N.S. Kroonenberg, Characterization of finite-dimensional $Z$-sets, Proc. Amer. Math. Soc. 83 (1977), 495--552.

6. J. van Mill, Infinite-dimension topology. Prerequisites and introduction, North-Holland, Amsterdam, 1989.

7. H. Toruńczyk, Concerning locally homotopy negligible sets and characterization of $l_2$-manifolds, Fund. Math. 101 (1978), 93--110.

8. R. Wong, A wild Cantor set in the Hilbert cube, Pacific J. Math. 24 (1968), 189–193.

	$Z_n$-sets and the disjoint $n$-cells property in products of ANR's
Author	T.O. Banakh, Kh.R. Trushchak tbanakh@franko.lviv.ua Department of Mechanics and Mathematics, Lviv National University, Universytetska 1, Lviv 79000, Ukraine
Abstract	It is shown that for $X,Y\in$ANR the product $A\times B$ of a $Z_n$-set $A\subset X$ and a $Z_m$-set $B\subset Y$ is a $Z_{n+m+1}$-set in $X\times Y$. This implies that the product $X\times Y$ of a~$\sigma Z_n$-space $X\in ANR$ and a~$\sigma Z_m$-space $Y\in ANR$ is a $\sigma Z_{n+m+1}$-space. Also we show that for every non-degenerate absolute retract $X$ and every $n$
Keywords	ANR spaces, product spaces, absolute neighborhood retracts, absolute retracts, dimension theory
DOI	doi:10.30970/ms.13.1.74-78
Reference	1. T.~Banakh, Some properties of the linear hull of the Erd\"os set in $l^2$, Bulletin PAN. Math. V.47 (1999), no.~4, 385--392. 2. T. Banakh, T. Radul, M. Zarichnyi, Absorbing sets in infinite-dimensional manifolds, VNTL Publishers, Lviv, 1996, 240 pp. 3. K. Borsuk, Theory of retracts, PWN, Warsaw, 1967. 4. T. Dobrowolski, The compact $Z$-set property in convex sets, Top. Appl. 23 (1986), 163--172. 5. N.S. Kroonenberg, Characterization of finite-dimensional $Z$-sets, Proc. Amer. Math. Soc. 83 (1977), 495--552. 6. J. van Mill, Infinite-dimension topology. Prerequisites and introduction, North-Holland, Amsterdam, 1989. 7. H. Toruńczyk, Concerning locally homotopy negligible sets and characterization of $l_2$-manifolds, Fund. Math. 101 (1978), 93--110. 8. R. Wong, A wild Cantor set in the Hilbert cube, Pacific J. Math. 24 (1968), 189–193.
Pages	74-78
Volume	13
Issue	1
Year	2000
Journal	Matematychni Studii
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