This article is cited in
2 papers
MATHEMATICS
On fully closed mappings of Fedorchuk compacta
S. P. Gul'koa,
A. V. Ivanovb a Tomsk State University,
Tomsk, Russian Federation
b Institute of Applied
Mathematics of Karelian Scientific Center of Russian Academy of Sciences, Petrozavodsk,
Russian Federation
Abstract:
An
$F$-compactum or a Fedorchuk compactum is a compact Hausdorff topological space that
admits a decomposition into a special fully ordered inverse spectrum with fully closed
neighboring projections.
$F$-compacta of spectral height
$3$ are exactly nonmetrizable compacta that
admit a fully closed mapping onto a metric compactum with metrizable fibers.
In this paper, it is proved that such a fully closed mapping for an
$F$-compactum
$X$ of spectral
height
$3$ is defined almost uniquely. Namely, nontrivial fibers of any two fully closed mapping of
$X$ into metric compacts with metrizable inverse images of points coincide everywhere, with a
possible exception of a countable family of elements.
Examples of
$F$-compacta of spectral height
$3$ are, for example, Aleksandrov’s "two arrows"
and the lexicographic square of the segment. It follows from the main result of this paper that
almost all non-trivial layers of any admissible fully closed mapping are colons that are glued
together under the standard projection of
$D$ onto the segment. Similarly, almost all nontrivial
fibers of any admissible fully closed mapping necessarily coincide with the "vertical segments" of
the lexicographic square.
UDC:
515.12 Received: 20.11.2017
DOI:
10.17223/19988621/50/1