On Closed Mappings of Sigma-Compact Spaces and Dimension
On Closed Mappings of Sigma-Compact Spaces and Dimension
We prove that if K is a remainder of the Hilbert space (i.e., K is the complement of the Hilbert space in its metrizable compactification) then every non-one-point closed image of K either contains a compact set with no transfinite dimension or contains compact sets of arbitrarily high inductive transfinite dimension ind. We construct also for each natural n a sigma-compact metrizable n-dimensional space whose image under any non-constant closed map has dimension at least n, and analogous examples for the transfinite dimension ind.
9 pages
transfinite small inductive dimension, Effros Borel space, closed mapping, Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets), Hilbert space, General Topology (math.GN), 57N20, 54D40, 54E40, 54F45, 54H05, Dimension theory in general topology, remainder, Topology of infinite-dimensional manifolds, FOS: Mathematics, Remainders in general topology, Special maps on metric spaces, Mathematics - General Topology
transfinite small inductive dimension, Effros Borel space, closed mapping, Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets), Hilbert space, General Topology (math.GN), 57N20, 54D40, 54E40, 54F45, 54H05, Dimension theory in general topology, remainder, Topology of infinite-dimensional manifolds, FOS: Mathematics, Remainders in general topology, Special maps on metric spaces, Mathematics - General Topology
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