\(\alpha_\beta\)-Connectedness as a characterization of connectedness

2018 ◽  
Vol 9 (2) ◽  
pp. 119-129 ◽  
Author(s):  
B. K. Tyagi ◽  
Manoj Bhardwaj ◽  
Sumit Singh
Keyword(s):  
Topological Space ◽  
Intermediate Value Theorem ◽  
New Class ◽  
Continuous Mappings ◽  
Intermediate Value ◽  
Connected Spaces

In this paper, a new class of \(\alpha_\beta\)-open sets in a topological space \(X\) is introduced which forms a topology on \(X\). The connectedness of this new topology on \(X\), called \(\alpha_\beta\)-connectedness, turns out to be equivalent to connectedness of \(X\) and hence also to \(\alpha\)-connectedness of \(X\). The \(\alpha_\beta\)-continuous and \(\alpha_\beta\)-irresolute mappings are defined and their relationship with other mappings such as continuous mappings and \(\alpha\)-continuous mappings are discussed. An intermediate value theorem is obtained. The hyperconnected spaces constitute a subclass of \(\alpha_\beta\)-connected spaces.

scholarly journals On Almost Continuous Mappings and Baire Spaces

1978 ◽  
Vol 21 (2) ◽  
pp. 183-186 ◽  
Author(s):  
Shwu-Yeng T. Lin ◽  
You-Feng Lin
Keyword(s):  
Topological Space ◽  
Dense Subset ◽  
Baire Space ◽  
Real Numbers ◽  
Range Space ◽  
Continuous Mappings ◽  
Almost Continuous

AbstractIt is proved, in particular, that a topological space X is a Baire space if and only if every real valued function f: X →R is almost continuous on a dense subset of X. In fact, in the above characterization of a Baire space, the range space R of real numbers may be generalized to any second countable, Hausdorfï space that contains infinitely many points.

scholarly journals The topology of θω-open sets

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5369-5377 ◽  
Author(s):  
Ghour Al ◽  
Bayan Irshedat
Keyword(s):  
Topological Space ◽  
Closure Operator ◽  
Sufficient Conditions ◽  
Topological Spaces ◽  
Product Theorem ◽  
Closure Operators ◽  
Separation Axiom ◽  
New Class

We define the ??-closure operator as a new topological operator. We show that ??-closure of a subset of a topological space is strictly between its usual closure and its ?-closure. Moreover, we give several sufficient conditions for the equivalence between ??-closure and usual closure operators, and between ??-closure and ?-closure operators. Also, we use the ??-closure operator to introduce ??-open sets as a new class of sets and we prove that this class of sets lies strictly between the class of open sets and the class of ?-open sets. We investigate ??-open sets, in particular, we obtain a product theorem and several mapping theorems. Moreover, we introduce ?-T2 as a new separation axiom by utilizing ?-open sets, we prove that the class of !-T2 is strictly between the class of T2 topological spaces and the class of T1 topological spaces. We study relationship between ?-T2 and ?-regularity. As main results of this paper, we give a characterization of ?-T2 via ??-closure and we give characterizations of ?-regularity via ??-closure and via ??-open sets.

scholarly journals ON An Infra-\(\alpha\)-Open Sets

2016 ◽  
Vol 4 (3) ◽  
pp. 12
Author(s):  
Hakeem Othman ◽  
Md.Hanif. Page
Keyword(s):  
General Topology ◽  
Connected Space ◽  
Basic Properties ◽  
New Class ◽  
Fundamental Properties ◽  
Open Set ◽  
Continuous Mappings

<p>In this paper, we define a new class of set in general topology called an infra- \(\alpha\) open set and we investigate fundamental properties by using this new class. The relation between infra-\(\alpha\)-open set and other topological sets are studied.</p><p>Moreover, In the light of this new definition, we also define some generalization of continuous mappings and discuss the relations between these new classes of mappings and other continuous mappings. Basic properties of these new mappings are studied and we apply these new classes to give characterization of connected space.</p>

scholarly journals The Reverse of the Intermediate Value Theorem in Some Topological Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Oussama Kabbouch ◽  
Mustapha Najmeddine
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Topological Spaces ◽  
Closed Graph ◽  
Intermediate Value Theorem ◽  
Hausdorff Topological Space ◽  
Intermediate Value

Any continuous function with values in a Hausdorff topological space has a closed graph and satisfies the property of intermediate value. However, the reverse implications are false, in general. In this article, we treat additional conditions on the function, and its graph for the reverse to be true.

scholarly journals Stirring our way to Sharkovsky's theorem

1997 ◽  
Vol 56 (3) ◽  
pp. 453-458
Author(s):  
Seth Patinkin
Keyword(s):  
Topological Space ◽  
Periodic Point ◽  
Real Line ◽  
Continuous Functions ◽  
Cycle Structure ◽  
Intermediate Value Theorem ◽  
The Real ◽  
Continuous Map ◽  
Sharkovsky's Theorem ◽  
Intermediate Value

The periodic-point or cycle structure of a continuous map of a topological space has been a subject of great interest since A.N. Sharkovsky completely explained the hierarchy of periodic orders of a continuous map f: R → R, where R is the real line. In this paper the topological idea of “stirring” is invoked in an effort to obtain a transparent proof of a generalisation of Sharkovsky's Theorem to continuous functions f: I → I where I is any interval. The stirring approach avoids all graph-theoretical and symbolic abstraction of the problem in favour of a more concrete intermediate-value-theorem-oriented analysis of cycles inside an interval.

Targeting IDH Mutations in AML: Wielding the Double-edged Sword of Differentiation

2020 ◽  
Vol 20 (7) ◽  
pp. 490-500 ◽  
Author(s):  
Justin S. Becker ◽  
Amir T. Fathi
Keyword(s):  
Genome Structure ◽  
Cellular Metabolism ◽  
Standard Of Care ◽  
Competitive Inhibitor ◽  
Driver Mutations ◽  
New Class ◽  
Regulatory Enzymes ◽  
Idh Mutations ◽  
Genomic Characterization

The genomic characterization of acute myeloid leukemia (AML) by DNA sequencing has illuminated subclasses of the disease, with distinct driver mutations, that might be responsive to targeted therapies. Approximately 15-23% of AML genomes harbor mutations in one of two isoforms of isocitrate dehydrogenase (IDH1 or IDH2). These enzymes are constitutive mediators of basic cellular metabolism, but their mutated forms in cancer synthesize an abnormal metabolite, 2- hydroxyglutarate, that in turn acts as a competitive inhibitor of multiple gene regulatory enzymes. As a result, leukemic IDH mutations cause changes in genome structure and gene activity, culminating in an arrest of normal myeloid differentiation. These discoveries have motivated the development of a new class of selective small molecules with the ability to inhibit the mutant IDH enzymes while sparing normal cellular metabolism. These agents have shown promising anti-leukemic activity in animal models and early clinical trials, and are now entering Phase 3 study. This review will focus on the growing preclinical and clinical data evaluating IDH inhibitors for the treatment of IDH-mutated AML. These data suggest that inducing cellular differentiation is central to the mechanism of clinical efficacy for IDH inhibitors, while also mediating toxicity for patients who experience IDH Differentiation Syndrome. Ongoing trials are studying the efficacy of IDH inhibitors in combination with other AML therapies, both to evaluate potential synergistic combinations as well as to identify the appropriate place for IDH inhibitors within existing standard-of-care regimens.

scholarly journals Characterization of the Antibiotic Compound No. 70 Produced byStreptomycessp. IMV-70

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Lyudmila P. Trenozhnikova ◽  
Almagul K. Khasenova ◽  
Assya S. Balgimbaeva ◽  
Galina B. Fedorova ◽  
Genrikh S. Katrukha ◽  
...  
Keyword(s):  
Optimal Growth ◽  
Growth Conditions ◽  
Culture Broth ◽  
Producer Strain ◽  
New Class ◽  
New Antibiotics ◽  
Actinomycete Strain ◽  
Main Component

We describe the actinomycete strain IMV-70 isolated from the soils of Kazakhstan, which produces potent antibiotics with high levels of antibacterial activity. After the research of its morphological, chemotaxonomic, and cultural characteristics, the strain with potential to be developed further as a novel class of antibiotics with chemotherapeutics potential was identified asStreptomycessp. IMV-70. In the process of fermentation, the strainStreptomycesspp. IMV-70 produces the antibiotic no. 70, which was isolated from the culture broth by extraction with organic solvents. Antibiotic compound no. 70 was purified and separated into individual components by HPLC, TLC, and column chromatography methods. The main component of the compound is the antibiotic 70-A, which was found to be identical to the peptolide etamycin A. Two other antibiotics 70-B and 70-C have never been described and therefore are new antibiotics. The physical-chemical and biological characteristics of these preparations were described and further researched. Determination of the optimal growth conditions to cultivate actinomycete-producer strain IMV-70 and development of methods to isolate, purify, and accumulate preparations of the new antibiotic no. 70 enable us to research further the potential of this new class of antibiotics.

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