scholarly journals A characterization of the uniform convergence points set of some convergent sequence of functions

2021 ◽  
Vol 71 (2) ◽  
pp. 423-428
Author(s):  
Olena Karlova
Keyword(s):  
Uniform Convergence ◽  
Convergent Sequence ◽  
Normal Space ◽  
Disjoint Sequence ◽  
Isolated Points ◽  
Perfectly Normal Space ◽  
Set Of Points ◽  
Sequence Of Functions ◽  
Perfectly Normal

Abstract We characterize the uniform convergence points set of a pointwisely convergent sequence of real-valued functions defined on a perfectly normal space. We prove that if X is a perfectly normal space which can be covered by a disjoint sequence of dense subsets and A ⊆ X, then A is the set of points of the uniform convergence for some convergent sequence (fn ) n∈ω of functions fn : X → ℝ if and only if A is Gδ -set which contains all isolated points of X. This result generalizes a theorem of Ján Borsík published in 2019.

scholarly journals PAIRS OF HAHN AND SEPARATELY CONTINUOUS FUNCTION

2021 ◽  
Vol 9 (1) ◽  
pp. 210-229
Author(s):  
O. Maslyuchenko ◽  
A. Kushnir
Keyword(s):  
Continuous Function ◽  
Lower Semicontinuous ◽  
Lower Semicontinuous Function ◽  
Normal Space ◽  
Semicontinuous Function ◽  
Upper Semicontinuous ◽  
Pseudocompact Space ◽  
Upper Semicontinuous Function ◽  
Perfectly Normal Space ◽  
Perfectly Normal

In this paper we continue the study of interconnections between separately continuous function which was started by V. K. Maslyuchenko. A pair (g, h) of functions on a topological space is called a pair of Hahn if g ≤ h, g is an upper semicontinuous function and h is a lower semicontinuous function. We say that a pair of Hahn (g, h) is generated by a function f, which depends on two variables, if the infimum of f and the supremum of f with respect to the second variable equals g and h respectively. We prove that for any perfectly normal space X and non-pseudocompact space Y every pair of Hahn on X is generated by a continuous function on X x Y . We also obtain that for any perfectly normal space X and for any space Y having non-scattered compactification any pair of Hahn on X is generated by a separately continuous function on X x Y .

Countably Compact Spaces and Martin's Axiom

1978 ◽  
Vol 30 (02) ◽  
pp. 243-249 ◽  
Author(s):  
William Weiss
Keyword(s):  
Set Theory ◽  
Normal Space ◽  
Topological Spaces ◽  
Compact Spaces ◽  
Countably Compact Space ◽  
Countably Compact ◽  
Countably Compact Spaces ◽  
Perfectly Normal Space ◽  
The Relationship ◽  
Perfectly Normal

The relationship between compact and countably compact topological spaces has been studied by many topologists. In particular an important question is: “What conditions will make a countably compact space compact?” Conditions which are “covering axioms” have been extensively studied. The best results of this type appear in [19]. We wish to examine countably compact spaces which are separable or perfectly normal. Recall that a space is perfect if and only if every closed subset is a Gδ, and that a space is perfectly normal if and only if it is both perfect and normal. We show that the following statement follows from MA +┐ CH and thus is consistent with the usual axioms of set theory: Every countably compact perfectly normal space is compact. This result is Theorem 3 and can be understood without reading much of what goes before.

scholarly journals Characterizations of Strongly Paracompact Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-5
Author(s):  
Xin Zhang
Keyword(s):  
Open Cover ◽  
Normal Space ◽  
Compact Spaces ◽  
Countably Paracompact ◽  
Star Countable ◽  
Perfectly Normal Space ◽  
Paracompact Spaces ◽  
Perfectly Normal

Characterizations of strongly compact spaces are given based on the existence of a star-countable open refinement for every increasing open cover. It is proved that a countably paracompact normal space (a perfectly normal space or a monotonically normal space) is strongly paracompact if and only if every increasing open cover of the space has a star-countable open refinement. Moreover, it is shown that a space is linearlyDprovided that every increasing open cover of the space has a point-countable open refinement.

scholarly journals (Strong) weak exhaustiveness and (strong uniform) continuity

Filomat ◽  
2010 ◽  
Vol 24 (4) ◽  
pp. 63-75 ◽  
Author(s):  
Agata Caserta ◽  
Maio Di ◽  
L'ubica Holá
Keyword(s):  
Uniform Convergence ◽  
Metric Spaces ◽  
Hausdorff Metric ◽  
Convergent Sequence ◽  
Uniform Continuity ◽  
Direct Image ◽  
Continuity Property ◽  
Property A ◽  
Sequence Of Functions

In this paper we continue, in the realm of metric spaces, the study of exhaustiveness and weak exhaustiveness at a point of a net of functions initiated by Gregoriades and Papanastassiou in 2008. We prove that exhaustiveness at every point of a net of pointwise convergent functions is equivalent to uniform convergence on compacta. We extend exhaustiveness-type properties to subsets. First, we introduce the notion of strong exhaustiveness at a subset B for sequences of functions and prove its equivalence with strong exhaustiveness at P0 (B) of the sequence of the direct image maps, where the hypersets are equipped with the Hausdorff metric. Furthermore, we show that the notion of strong-weak exhaustiveness at a subset is the proper tool to investigate when the limit of a pointwise convergent sequence of functions fulfills the strong uniform continuity property, a new pregnant form of uniform continuity discovered by Beer and Levi in 2009.

An example of a hereditarily normal topologically finite space, which is topologically infinite relative to the class of all its proper F σ-subspaces

2019 ◽  
Vol 26 (2) ◽  
pp. 315-319
Author(s):  
Ivane Tsereteli
Keyword(s):  
Normal Space ◽  
Finite Space ◽  
Perfectly Normal Space ◽  
Open Subspace ◽  
Perfectly Normal

Abstract A (Hausdorf) hereditarily normal (not perfectly normal) space X is constructed, which has the following properties: (a) there exists a proper open subspace of X which is homeomorphic to the whole X (i.e., the space X is topologically infinite); (b) the space is homeomorphic to none of its proper {F_{\sigma}} -subspaces (i.e., the space X is topologically finite relative to the class of all its proper {F_{\sigma}} -subspaces).

scholarly journals Local minima of the Gauss curvature of a minimal surface

1991 ◽  
Vol 44 (3) ◽  
pp. 397-404
Author(s):  
Shinji Yamashita
Keyword(s):  
Minimal Surface ◽  
Gauss Curvature ◽  
Gauss Map ◽  
Local Minima ◽  
Jordan Curves ◽  
The Third ◽  
Isolated Points ◽  
Set Of Points

Let D be a domain in the complex ω-plane and let x: D → R3 be a regular minimal surface. Let M(K) be the set of points ω0 ∈ D where the Gauss curvature K attains local minima: K(ω0) ≤ K(ω) for |ω – ω0| < δ(ω0), δ(ω0) < 0. The components of M(K) are of three types: isolated points; simple analytic arcs terminating nowhere in D; analytic Jordan curves in D. Components of the third type are related to the Gauss map.

On the Characterization of the Set of Points of λ-Continuity

1923 ◽  
Vol 25 (2) ◽  
pp. 118
Author(s):  
Henry Blumberg
Keyword(s):  
Set Of Points
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