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The approximation of compacta by finite T0-spaces
Abstract
It has long been known that compact Hausdorff
spaces can be approximated using finite T0-spaces,
and that many can be represented as inverse limits of polyhedra. Here we study
the relationship between these two types of representation. In Section 4, we
define the concept of a calming map and show that the Hausdorff reflection of
the limit of an inverse sequence of finite T0-spaces
and calming maps is the inverse limit of their corresponding polytopes and
piecewise linear maps. Thus each k-dimensional
metric compactum (respectively, continuum) can be characterized as the
Hausdorff reflection of the limit of an inverse sequence with calming bonding
maps of finite (respectively, connected) T0-spaces
whose dimension is k; an
infinite-dimensional version of this is also found.
Mathematics Subject Classification (2000): Primary 54E45, 54E55, 54F15, 54D30
Key words: Specializatoin order, Alexandroff dimension, (small) inductive dimension, simplicialization, separating map,
Hausdorff reflection, chaining map, calming map, barycentric subdivision, piecewise linear map, simplicial map, complex map, complex, polytope.
Quaestiones Mathematicae 26(2003), 371–383
spaces can be approximated using finite T0-spaces,
and that many can be represented as inverse limits of polyhedra. Here we study
the relationship between these two types of representation. In Section 4, we
define the concept of a calming map and show that the Hausdorff reflection of
the limit of an inverse sequence of finite T0-spaces
and calming maps is the inverse limit of their corresponding polytopes and
piecewise linear maps. Thus each k-dimensional
metric compactum (respectively, continuum) can be characterized as the
Hausdorff reflection of the limit of an inverse sequence with calming bonding
maps of finite (respectively, connected) T0-spaces
whose dimension is k; an
infinite-dimensional version of this is also found.
Mathematics Subject Classification (2000): Primary 54E45, 54E55, 54F15, 54D30
Key words: Specializatoin order, Alexandroff dimension, (small) inductive dimension, simplicialization, separating map,
Hausdorff reflection, chaining map, calming map, barycentric subdivision, piecewise linear map, simplicial map, complex map, complex, polytope.
Quaestiones Mathematicae 26(2003), 371–383
