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A characterization of ω-limit sets for piecewise monotone maps of the interval

Andrew D. Barwell (2010)

Fundamenta Mathematicae

For a piecewise monotone map f on a compact interval I, we characterize the ω-limit sets that are bounded away from the post-critical points of f. If the pre-critical points of f are dense, for example when f is locally eventually onto, and Λ ⊂ I is closed, invariant and contains no post-critical point, then Λ is the ω-limit set of a point in I if and only if Λ is internally chain transitive in the sense of Hirsch, Smith and Zhao; the proof relies upon symbolic dynamics. By identifying points of...

A nice class extracted from C p -theory

Vladimir Vladimirovich Tkachuk (2005)

Commentationes Mathematicae Universitatis Carolinae

We study systematically a class of spaces introduced by Sokolov and call them Sokolov spaces. Their importance can be seen from the fact that every Corson compact space is a Sokolov space. We show that every Sokolov space is collectionwise normal, ω -stable and ω -monolithic. It is also established that any Sokolov compact space X is Fréchet-Urysohn and the space C p ( X ) is Lindelöf. We prove that any Sokolov space with a G δ -diagonal has a countable network and obtain some cardinality restrictions on subsets...

A note on inverse limits of continuous images of arcs.

Ivan Loncar (1999)

Publicacions Matemàtiques

The main purpose of this paper is to prove some theorems concerning inverse systems and limits of continuous images of arcs. In particular, we shall prove that if X = {Xa, pab, A} is an inverse system of continuous images of arcs with monotone bonding mappings such that cf (card (A)) ≠ w1, then X = lim X is a continuous image of an arc if and only if each proper subsystem {Xa, pab, B} of X with cf(card (B)) = w1 has the limit which is a continuous image of an arc (Theorem 18).

A note on pseudobounded paratopological groups

Fucai Lin, Shou Lin, Iván Sánchez (2014)

Topological Algebra and its Applications

Let G be a paratopological group. Then G is said to be pseudobounded (resp. ω-pseudobounded) if for every neighbourhood V of the identity e in G, there exists a natural number n such that G = Vn (resp.we have G = ∪ n∈N Vn). We show that every feebly compact (2-pseudocompact) pseudobounded (ω-pseudobounded) premeager paratopological group is a topological group. Also,we prove that if G is a totally ω-pseudobounded paratopological group such that G is a Lusin space, then is G a topological group....

A revised closed graph theorem for quasi-Suslin spaces

Juan Carlos Ferrando, J. Kąkol, M. Lopez Pellicer (2009)

Czechoslovak Mathematical Journal

Some results about the continuity of special linear maps between F -spaces recently obtained by Drewnowski have motivated us to revise a closed graph theorem for quasi-Suslin spaces due to Valdivia. We extend Valdivia’s theorem by showing that a linear map with closed graph from a Baire tvs into a tvs admitting a relatively countably compact resolution is continuous. This also applies to extend a result of De Wilde and Sunyach. A topological space X is said to have a (relatively countably) compact...

A topological dichotomy with applications to complex analysis

Iosif Pinelis (2015)

Colloquium Mathematicae

Let X be a compact topological space, and let D be a subset of X. Let Y be a Hausdorff topological space. Let f be a continuous map of the closure of D to Y such that f(D) is open. Let E be any connected subset of the complement (to Y) of the image f(∂D) of the boundary ∂D of D. Then f(D) either contains E or is contained in the complement of E. Applications of this dichotomy principle are given, in particular for holomorphic maps, including maximum and minimum modulus principles,...

Algebras and spaces of dense constancies

Angelo Bella, Jorge Martinez, Scott D. Woodward (2001)

Czechoslovak Mathematical Journal

A DC-space (or space of dense constancies) is a Tychonoff space X such that for each f ∈ C ( X ) there is a family of open sets { U i i ∈ I } , the union of which is dense in X , such that f , restricted to each U i , is constant. A number of characterizations of DC-spaces are given, which lead to an algebraic generalization of the concept, which, in turn, permits analysis of DC-spaces in the language of archimedean f -algebras. One is led naturally to the notion of an almost DC-space (in which the densely constant functions...

Almost g ˜ α -closed functions and separation axioms

O. Ravi, S. Ganesan, R. Latha (2012)

Mathematica Bohemica

We introduce a new class of functions called almost g ˜ α -closed and use the functions to improve several preservation theorems of normality and regularity and also their generalizations. The main result of the paper is that normality and weak normality are preserved under almost g ˜ α -closed continuous surjections.

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