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In this paper, we discuss certain
networks on paratopological (or
topological) groups and give positive
or negative answers to the questions
in [Lin2013]. We also prove that a
non-locally compact, -gentle
paratopological group is metrizable if
its remainder (in the Hausdorff
compactification) is
a Fréchet-Urysohn space with a
point-countable cs*-network, which
improves some theorems in
[Liu C., Metrizability of paratopological
semitopological groups,
Topology Appl. 159 (2012), 1415–1420],
[Liu...
We show that a Tychonoff space is the perfect pre-image of a cofinally complete metric space if and only if it is paracompact and cofinally Čech complete. Further properties of these spaces are discussed. In particular, cofinal Čech completeness is preserved both by perfect mappings and by continuous open mappings.
We provide a necessary and sufficient condition for the Higson compactification to be perfect for the noncompact, locally connected, proper metric spaces. We also discuss perfectness of the Smirnov compactification.
We prove that in some families of planar rational compacta there are no universal elements.
Let be a continuous map with the specification property on a compact metric space . We introduce the notion of the maximal Birkhoff average oscillation, which is the “worst” divergence point for Birkhoff average. By constructing a kind of dynamical Moran subset, we prove that the set of points having maximal Birkhoff average oscillation is residual if it is not empty. As applications, we present the corresponding results for the Birkhoff averages for continuous functions on a repeller and locally...
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