scholarly journals Algebras with actions and automata

1982 ◽  
Vol 5 (1) ◽  
pp. 61-85
Author(s):  
W. Kühnel ◽  
M. Pfender ◽  
J. Meseguer ◽  
I. Sols
Keyword(s):  
Axiomatic Approach ◽  
Structure Theory ◽  
Topological Spaces ◽  
Algebraic Structures ◽  
Hausdorff Spaces ◽  
Left Action ◽  
Common Structure ◽  
The One ◽  
Cartesian Closed ◽  
In The Beginning

In the present paper we want to give a common structure theory of left action, group operations,R-modules and automata of different types defined over various kinds of carrier objects: sets, graphs, presheaves, sheaves, topological spaces (in particular: compactly generated Hausdorff spaces). The first section gives an axiomatic approach to algebraic structures relative to a base categoryB, slightly more powerful than that of monadic (tripleable) functors. In section2we generalize Lawveres functorial semantics to many-sorted algebras over cartesian closed categories. In section3we treat the structures mentioned in the beginning as many-sorted algebras with fixed “scalar” or “input” object and show that they still have an algebraic (or monadic) forgetful functor (theorem 3.3) and hence the general theory of algebraic structures applies. These structures were usually treated as one-sorted in the Lawvere-setting, the action being expressed by a family of unary operations indexed over the scalars. But this approach cannot, as the one developed here, describe continuity of the action (more general: the action to be aB-morphism), which is essential for the structures mentioned above, e.g. modules for a sheaf of rings or topological automata. Finally we discuss consequences of theorem 3.3 for the structure theory of various types of automata. The particular case of algebras with fixed “natural numbers object” has been studied by the authors in [23].

Essentials of General Topology

2021 ◽  
pp. 191-244
Author(s):  
Adel N. Boules
Keyword(s):  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Compact Spaces ◽  
Section 8 ◽  
Finite Products ◽  
The One ◽  
One Point Compactification ◽  
Locally Compact Spaces ◽  
Metrization Theorem

The first eight sections of this chapter constitute its core and are generally parallel to the leading sections of chapter 4. Most of the sections are brief and emphasize the nonmetric aspects of topology. Among the topics treated are normality, regularity, and second countability. The proof of Tychonoff’s theorem for finite products appears in section 8. The section on locally compact spaces is the transition between the core of the chapter and the more advanced sections on metrization, compactification, and the product of infinitely many spaces. The highlights include the one-point compactification, the Urysohn metrization theorem, and Tychonoff’s theorem. Little subsequent material is based on the last three sections. At various points in the book, it is explained how results stated for the metric case can be extended to topological spaces, especially locally compact Hausdorff spaces. Some such results are developed in the exercises.

scholarly journals BOUNDING THE MINIMUM ORDER OF Aπ-BASE

2011 ◽  
Vol 83 (2) ◽  
pp. 321-328
Author(s):  
MARÍA MUÑOZ
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Minimum Order ◽  
Compact Spaces ◽  
Cardinal Invariants ◽  
Open Set ◽  
Open Questions ◽  
The Family ◽  
The One

AbstractLetXbe a topological space. A family ℬ of nonempty open sets inXis called aπ-base ofXif for each open setUinXthere existsB∈ℬ such thatB⊂U. The order of aπ-base ℬ at a pointxis the cardinality of the family ℬx={B∈ℬ:x∈B} and the order of theπ-base ℬ is the supremum of the orders of ℬ at each pointx∈X. A classical theorem of Shapirovskiĭ [‘Special types of embeddings in Tychonoff cubes’, in:Subspaces of Σ-Products and Cardinal Invariants, Topology, Coll. Math. Soc. J. Bolyai, 23 (North-Holland, Amsterdam, 1980), pp. 1055–1086; ‘Cardinal invariants in compact Hausdorff spaces’,Amer. Math. Soc. Transl.134(1987), 93–118] establishes that the minimum order of aπ-base is bounded by the tightness of the space when the space is compact. Since then, there have been many attempts at improving the result. Finally, in [‘The projectiveπ-character bounds the order of aπ-base’,Proc. Amer. Math. Soc.136(2008), 2979–2984], Juhász and Szentmiklóssy proved that the minimum order of aπ-base is bounded by the ‘projectiveπ-character’ of the space for any topological space (not only for compact spaces), improving Shapirovskiĭ’s theorem. The projectiveπ-character is in some sense an ‘external’ cardinal function. Our purpose in this paper is, on the one hand, to give bounds of the projectiveπ-character using ‘internal’ topological properties of the subspaces on compact spaces. On the other hand, we give a bound on the minimum order of aπ-base using other cardinal functions in the frame of general topological spaces. Open questions are posed.

Extensions of Topological Spaces

1964 ◽  
Vol 7 (1) ◽  
pp. 1-22 ◽  
Author(s):  
Bernhard Banaschewski
Keyword(s):  
Complex Number ◽  
Large Body ◽  
General Topology ◽  
General Question ◽  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Completely Regular ◽  
Wide Range ◽  
Extension Spaces ◽  
The One

The undertaking of constructing spaces which contain a given space as a subspace is by no means new: the extension of the complex number plane to the complex number sphere by the addition of the one point at infinity, the extension of the real line by adjoining the two infinities ∞ and -∞, and the construction of the space of real numbers from that of the rationals by means of Cauchy sequences or Dedekind cuts are 19th Century examples of this very thing. However, only the advent of general topology made it possible to raise the general question of space extensions. It appears that the first study of problems in this area was carried out by Alexandroff and Urysohn in the early twenties [l]. Another mile stone in the history of the subject was the 1929 paper by Tychonoff in which the product theorem for compact spaces is proved and used to identify the completely regular Hausdorff spaces as precisely those spaces which can be imbedded in a compact Hausdorff space [33]. During the same period, work on certain specific extension problems was done by Freudenthal [17] and Zippin [35]. However, the first large body of systematic theory, used for the investigation of a wide range of extension problems, was presented by Stone [31] in 1937. There, one also finds the remark that "one of the interesting and difficult problems of general topology is the study of all extensions of a given space", and it appears that Stone' s own work must have convinced many others of the truth of this observation, for since that time there has been a steady succession of papers in this field. But apart from that, the study of extension spaces clearly has a very particular attraction for some mathematicians.

scholarly journals Topological and limit-space subcategories of countably-based equilogical spaces

2002 ◽  
Vol 12 (6) ◽  
pp. 739-770 ◽  
Author(s):  
MATÍAS MENNI ◽  
ALEX SIMPSON
Keyword(s):  
Large Class ◽  
Full Subcategory ◽  
The Other ◽  
Topological Spaces ◽  
Limit Space ◽  
Quotient Spaces ◽  
Cartesian Closed Categories ◽  
The One ◽  
Cartesian Closed ◽  
Sequential Spaces

There are two main approaches to obtaining ‘topological’ cartesian-closed categories. Under one approach, one restricts to a full subcategory of topological spaces that happens to be cartesian closed – for example, the category of sequential spaces. Under the other, one generalises the notion of space – for example, to Scott's notion of equilogical space. In this paper, we show that the two approaches are equivalent for a large class of objects. We first observe that the category of countably based equilogical spaces has, in a precisely defined sense, a largest full subcategory that can be simultaneously viewed as a full subcategory of topological spaces. In fact, this category turns out to be equivalent to the category of all quotient spaces of countably based topological spaces. We show that the category is bicartesian closed with its structure inherited, on the one hand, from the category of sequential spaces, and, on the other, from the category of equilogical spaces. We also show that the category of countably based equilogical spaces has a larger full subcategory that can be simultaneously viewed as a full subcategory of limit spaces. This full subcategory is locally cartesian closed and the embeddings into limit spaces and countably based equilogical spaces preserve this structure. We observe that it seems essential to go beyond the realm of topological spaces to achieve this result.

scholarly journals Two semigroups of continuous relations

1977 ◽  
Vol 23 (1) ◽  
pp. 46-58 ◽  
Author(s):  
A. R. Bednarek ◽  
Eugene M. Norris
Keyword(s):  
Large Class ◽  
Algebraic Structure ◽  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Completely Regular ◽  
Topological Semigroups ◽  
Stone Type

SynopsisIn this paper we define two semigroups of continuous relations on topological spaces and determine a large class of spaces for which Banach-Stone type theorems hold, i.e. spaces for which isomorphism of the semigroups implies homeomorphism of the spaces. This class includes all 0-dimensional Hausdorff spaces and all those completely regular Hausdorff spaces which contain an arc; indeed all of K. D. Magill's S*-spaces are included. Some of the algebraic structure of the semigroup of all continuous relations is elucidated and a method for producing examples of topological semigroups of relations is discussed.

Il triteismo nel VI secolo: la fase arcaica (557-567)

Augustinianum ◽  
2020 ◽  
Vol 60 (1) ◽  
pp. 189-207
Author(s):  
Carlo dell’Osso ◽  
Keyword(s):  
Sixth Century ◽  
Imperial Capital ◽  
Doctrine Of The Trinity ◽  
The One ◽  
In The Beginning ◽  
The Trinity

The Tritheism of the sixth century has not been widely studied. John Philoponus, the greatest exponent of the theory, developed the idea by applying Aristotelian realism to the doctrine of the Trinity and concluded that in the Trinity there are three hypostases and three natures, whence comes the name for those who hold this position: “Tri-theists,” since they divide the one nature and substance of God into three. This article sheds light on the earliest stage of the development of Tritheism beginning in the year 557, when we can date the first appearance of John Askotzanges in the sources, and goes up until the first Syndocticon, the agreement reached between the Tritheists and the Theodosians at Constantinople in the beginning of the year 567. After the death of Theodosius in 566, Tritheism no longer remained merely a local reality in Constantinople but spilled over the confines of the Imperial capital and spread throughout the East, especially in Egypt.

scholarly journals Study of a Forwarding Chain in the Category of Topological Spaces between T0 and T2 with respect to One Point Compactification Operator

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Fatemah Ayatollah Zadeh Shirazi ◽  
Meysam Miralaei ◽  
Fariba Zeinal Zadeh Farhadi
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
The One ◽  
One Point Compactification

In the following text, we want to study the behavior of one point compactification operator in the chain Ξ := {Metrizable, Normal, T2, KC, SC, US, T1, TD, TUD, T0, Top} of subcategories of category of topological spaces, Top (where we denote the subcategory of Top, containing all topological spaces with property P , simply by P). Actually we want to know, for P∈Ξ and X∈P, the one point compactification of topological space X belongs to which elements of Ξ. Finally we find out that the chain {Metrizable, T2, KC, SC, US, T1, TD, TUD, T0, Top} is a forwarding chain with respect to one point compactification operator.

scholarly journals Multidrug-resistant bacteria isolated from surgical site of dogs, surgeon's hands and operating room in a veterinary teaching hospital in Brazil

2021 ◽  
Author(s):  
Mareliza Possa de Menezes ◽  
Mariana Borzi ◽  
Mayara Ruaro ◽  
Marita Cardozo ◽  
Fernando Ávila ◽  
...  
Keyword(s):  
Multidrug Resistant ◽  
High Rate ◽  
Resistant Bacteria ◽  
Coagulase Negative Staphylococci ◽  
Gram Negative ◽  
Source Of Infection ◽  
The One ◽  
Clean Contaminated ◽  
In The Beginning ◽  
Coagulase Positive Staphylococci

Abstract The aim of this study was to evaluate the prevalence and antimicrobial resistance profile of Gram-positive cocci and Gram-negative bacilli isolated from the surgical environment. All samples were collected during the intraoperative period of clean/clean-contaminated (G1) and contaminated (G2) surgery. A total of 150 samples were collected from the surgical wound in the beginning (n = 30) and end (n = 30) of the procedure, surgeon’s hands before (n = 30) and after (n = 30) antisepsis and the surgical environment (n = 30). Forty-three isolates with morphological and biochemical characteristics of Staphylococcus spp. and 13 of Gram-negative bacilli were obtained. Coagulase-negative staphylococci (85.71% [18/21]), coagulase-positive staphylococci (9.52% [2/21]) and Pseudomonas spp. (47.52% [1/21]) in G1, and coagulase-negative staphylococci (40% [14/35]), coagulase-positive staphylococci (20% [7/35]), Proteus spp. (17.14% [6/35]), E. coli (8.57% [3/35]), Pseudomonas spp. (2.86% [1/35]) and Salmonella spp. (2.86 [1/35]) in G2 were more frequently isolated, and a high incidence of multidrug resistance was observed in coagulase-negative staphylococci (87.5% [28/32]), coagulase-positive staphylococci (100% [11/11]) and Gram-negative bacilli (76.92% [10/13]). Methicillin-resistant Staphylococcus spp. accounted for 83.72% (36/43) of the Staphylococcus strains. Gram-negative bacilli cefotaxime-resistance constituted 81.82% (9/11) and imipenem resistance constituted 53.85% (7/13). The high rate of resistance of commensal bacteria found in our study is worrying. Coagulase-negative staphylococci are community pathogens related to nosocomial infections in human and veterinary hospitals, their presence in healthy patients and in veterinary professionals represent an important source of infection in the one health context. Continuous surveillance and application of antimicrobial stewardship programs are essential in the fight against this threat.

scholarly journals Obrona integralnego człowieczeństwa Chrystusa przeciw apolinaryzmowi w dziełach Epifaniusza z Salaminy

Vox Patrum ◽  
2018 ◽  
Vol 68 ◽  
pp. 253-269
Author(s):  
Roland Marcin Pancerz
Keyword(s):  
Human Nature ◽  
Human Mind ◽  
Church Fathers ◽  
Hypostatic Union ◽  
Apostolic Fathers ◽  
Communicatio Idiomatum ◽  
The One ◽  
In The Beginning ◽  
The Church ◽  
Technical Terms

Epiphanius of Salamis was one of the Church Fathers, who reacted resolutely against incorrect Christology of Apollinaris of Laodicea. The latter asserted that the divine Logos took the place of Christ’s human mind (noàj). In the beginning, the bishop of Salamis tackled the problem of Christ’s human body, since – as he told himself – followers of Apollinaris, that arrived in Cyprus, put about incorrect doctrine on the Saviour’s body. Among other things, they asserted it was consub­stantial with his godhead. Beyond doubt, this idea constituted a deformation of the original thought of Apollinaris. Anyway, Epiphanius opposing that error took up again expressions, which had been employed before by the Apostolic Fathers and Apologists in the fight against Docetism. Besides, Epiphanius told that some followers of Apollinaris denied the exi­stence of Christ’s human soul (yuc»). Also in this matter, in all probability, we come across a deformation of the original doctrine of the bishop of Laodicea. A real controversy with Apollinaris was the defence of the human mind of the Sa­viour. Epiphanius emphasized that He becoming man took all components of hu­man nature: “body, soul, mind and everything that man is”, in accordance with the axiom “What is not assumed is not saved” (Quod non assumptum, non sanatum). A proof of the integrity of human nature was the reasonable human feelings the Saviour experienced (hunger, tiredness, sorrow, anxiety) as well as knowledge he had to gain partly from experience, which was witnessed by Luke 2, 52. In the lat­ter question, the bishop of Salamis was a forerunner of contemporary Christology. The fact that Epiphanius admitted a complete human nature in Christ didn’t bring dividing the incarnate Logos into two persons. Although the bishop of Sa­lamis didn’t use technical terms for the one person of Jesus Christ, he outlined nonetheless the idea of the hypostatic union in his own words, as well as through employing the rule of the communicatio idiomatum. The ontological union of the divine Logos with his human nature assured Christ’s holiness, too.

scholarly journals CONTINUOUS ORDER REPRESENTABILITY PROPERTIES OF TOPOLOGICAL SPACES AND ALGEBRAIC STRUCTURES

2012 ◽  
Vol 49 (3) ◽  
pp. 449-473 ◽  
Author(s):  
Maria Jesus Campion ◽  
Juan Carlos Candeal ◽  
Esteban Indurain ◽  
Ghanshyam Bhagvandas Mehta
Keyword(s):  
Topological Spaces ◽  
Algebraic Structures
Sign in / Sign up

Export Citation Format

Share Document