We study topological properties of the escaping endpoints and fast escaping endpoints of the Julia set of complex exponential
$\exp (z)+a$
when
$a\in (-\infty ,-1)$
. We show neither space is homeomorphic to the whole set of endpoints. This follows from a general result stating that for every transcendental entire function
$f$
, the escaping Julia set
$I(f)\cap J(f)$
is first category.
A connected topological space is said to be widely-connected if each of its non-degenerate connected subsets is dense in the entire space. The object of this paper is the construction of widely-connected subsets of the plane. We give a completely metrizable example that answers a question of Paul Erdős and Howard Cook, while a similar example answers a question of Jerzy Mioduszewski.
The following is an open problem in topology: Determine whether the Stone-Čech compactification of a widely-connected space is necessarily an indecomposable continuum. Herein we describe properties of X that are necessary and sufficient in order for βX to be indecomposable. We show that indecomposability and irreducibility are equivalent properties in compactifications of widely-connected separable metric spaces, leading to some equivalent formulations of the open problem. We also construct a widely-connected subset of Euclidean 3-space which is contained in a composant of each of its compactifications. The example answers a question of Jerzy Mioduszewski.
We construct two connected plane sets which can be embedded into rational curves. The first is a biconnected set with a dispersion point. It answers a question of Joachim Grispolakis. The second is indecomposable. Both examples are completely metrizable.2010 Mathematics Subject Classification. 54F45, 54F15, 54D35, 54G20.
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