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Hilbert cube; absorbing system; $F_\sigma$; $F_{\sigma \delta}$; capset; Peano continuum; hyperspace; hyperspace of subcontinua; covering dimension; cohomological dimension
Let $\Cal D$ denote a true dimension function, i.e., a dimension function such that $\Cal D(\Bbb R^n) = n$ for all $n$. For a space $X$, we denote the hyperspace consisting of all compact connected, non-empty subsets by $C(X)$. If $X$ is a countable infinite product of non-degenerate Peano continua, then the sequence $(\Cal D_{\geq n}(C(X)))_{n=2}^\infty$ is $F_\sigma$-absorbing in $C(X)$. As a consequence, there is a homeomorphism $h: C(X)\rightarrow Q^\infty$ such that for all $n$, $h[\{A \in C(X) : \Cal D(A) \geq n+1\}] = B^n \times Q \times Q \times \dots $, where $B$ denotes the pseudo boundary of the Hilbert cube $Q$. It follows that if $X$ is a countable infinite product of non-degenerate Peano continua then $\Cal D_{\geq n}(C(X))$ is an $F_\sigma$-absorber (capset) for $C(X)$, for every $n \geq 2$. Let $\operatorname{dim}$ denote covering dimension. It is known that there is an example of an everywhere infinite dimensional Peano continuum $X$ that contains arbitrary large $n$-cubes, such that for every $k \in \Bbb N$, the sequence $(\operatorname{dim}_{\geq n}(C(X^k)))_{n=2}^\infty$ is not $F_\sigma$-absorbing in $C(X^k)$. So our result is in some sense the best possible.
[1] Bessaga C., Pełczyński A.: Selected topics in infinite-dimensional topology. PWN, Warszawa, 1975.
[2] Bing R.H.: Partitioning a set. Bull. Amer. Math. Soc. 55 (1949), 1101-1110. MR 0035429 | Zbl 0036.11702
[3] Curtis D.W.: Boundary sets in the Hilbert cube. Top. Appl. 20 (1985), 201-221. MR 0804034 | Zbl 0575.57008
[4] Curtis D.W., Nhu N.T.: Hyperspaces of finite subsets which are homeomorphic to $\aleph _0$-dimensional linear metric space. Top. Appl. 19 (1985), 251-260. MR 0794488
[5] Curtis D.W., Michael M.: Boundary sets for growth hyperspaces. Top. Appl. 25 (1987), 269-283. MR 0889871 | Zbl 0627.54004
[6] Gladdines H., van Mill J.: Hyperspaces of infinite-dimensional compacta. Comp. Math. 88 (1993), 143-153. MR 1237918 | Zbl 0830.57013
[7] Gladdines H.: $F_\sigma $-absorbing sequences in hyperspaces of compact sets. Bull. Pol. Ac. Sci. vol 40 (3) (1992). MR 1401869
[8] Gladdines H., Baars J., van Mill J.: Absorbing systems in infinite-dimensional manifolds. Topology Appl. 50 (1993), 147-182. MR 1217483 | Zbl 0794.57005
[9] Dobrowolski T., Rubin L.R.: The hyperspace of infinite-dimensional compacta for covering and cohomological dimension are homeomorphic. preprint, to appear. MR 1267500
[10] Dijkstra J.J., van Mill J., Mogilski J.: The space of infinite-dimensional compact spaces and other topological copies of $(\ell _f^2)^ømega $. Pac. J. Math. 152 (1992), 255-273. MR 1141795
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