2,178 research outputs found
Quasi-pseudo-metrization of topological preordered spaces
We establish that every second countable completely regularly preordered
space (E,T,\leq) is quasi-pseudo-metrizable, in the sense that there is a
quasi-pseudo-metric p on E for which the pseudo-metric p\veep^-1 induces T and
the graph of \leq is exactly the set {(x,y): p(x,y)=0}. In the ordered case it
is proved that these spaces can be characterized as being order homeomorphic to
subspaces of the ordered Hilbert cube. The connection with
quasi-pseudo-metrization results obtained in bitopology is clarified. In
particular, strictly quasi-pseudometrizable ordered spaces are characterized as
being order homeomorphic to order subspaces of the ordered Hilbert cube.Comment: Latex2e, 20 pages. v2: minor changes in the proof of theorem 2.
A Birkhoff type transitivity theorem for non-separable completely metrizable spaces with applications to Linear Dynamics
In this note we prove a Birkhoff type transitivity theorem for continuous
maps acting on non-separable completely metrizable spaces and we give some
applications for dynamics of bounded linear operators acting on complex
Fr\'{e}chet spaces. Among them we show that any positive power and any
unimodular multiple of a topologically transitive linear operator is
topologically transitive, generalizing similar results of S.I. Ansari and F.
Le\'{o}n-Saavedra V. M\"{u}ller for hypercyclic operators.Comment: Several changes concerning the presentation of the paper; title
changed; 12 page
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