Minimal Topological Spaces(')
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Filtering Germs: Groupoids Associated to Inverse Semigroups
FILTERING GERMS: GROUPOIDS ASSOCIATED TO INVERSE SEMIGROUPS BECKY ARMSTRONG, LISA ORLOFF CLARK, ASTRID AN HUEF, MALCOLM JONES, AND YING-FEN LIN Abstract. We investigate various groupoids associated to an arbitrary inverse semigroup with zero. We show that the groupoid of filters with respect to the natural partial order is isomorphic to the groupoid of germs arising from the standard action of the inverse semigroup on the space of idempotent filters. We also investigate the restriction of this isomorphism to the groupoid of tight filters and to the groupoid of ultrafilters. 1. Introduction An inverse semigroup is a set S endowed with an associative binary operation such that for each a 2 S, there is a unique a∗ 2 S, called the inverse of a, satisfying aa∗a = a and a∗aa∗ = a∗: The study of ´etalegroupoids associated to inverse semigroups was initiated by Renault [Ren80, Remark III.2.4]. We consider two well known groupoid constructions: the filter approach and the germ approach, and we show that the two approaches yield isomorphic groupoids. Every inverse semigroup has a natural partial order, and a filter is a nonempty down-directed up-set with respect to this order. The filter approach to groupoid construction first appeared in [Len08], and was later simplified in [LMS13]. Work in this area is ongoing; see for instance, [Bic21, BC20, Cas20]. Every inverse semigroup acts on the filters of its subsemigroup of idempotents. The groupoid of germs associated to an inverse semigroup encodes this action. Paterson pioneered the germ approach in [Pat99] with the introduction of the universal groupoid of an inverse semigroup. -
Relations on Semigroups
International Journal for Research in Engineering Application & Management (IJREAM) ISSN : 2454-9150 Vol-04, Issue-09, Dec 2018 Relations on Semigroups 1D.D.Padma Priya, 2G.Shobhalatha, 3U.Nagireddy, 4R.Bhuvana Vijaya 1 Sr.Assistant Professor, Department of Mathematics, New Horizon College Of Engineering, Bangalore, India, Research scholar, Department of Mathematics, JNTUA- Anantapuram [email protected] 2Professor, Department of Mathematics, SKU-Anantapuram, India, [email protected] 3Assistant Professor, Rayalaseema University, Kurnool, India, [email protected] 4Associate Professor, Department of Mathematics, JNTUA- Anantapuram, India, [email protected] Abstract: Equivalence relations play a vital role in the study of quotient structures of different algebraic structures. Semigroups being one of the algebraic structures are sets with associative binary operation defined on them. Semigroup theory is one of such subject to determine and analyze equivalence relations in the sense that it could be easily understood. This paper contains the quotient structures of semigroups by extending equivalence relations as congruences. We define different types of relations on the semigroups and prove they are equivalence, partial order, congruence or weakly separative congruence relations. Keywords: Semigroup, binary relation, Equivalence and congruence relations. I. INTRODUCTION [1,2,3 and 4] Algebraic structures play a prominent role in mathematics with wide range of applications in science and engineering. A semigroup -
1.1.5 Hausdorff Spaces
1. Preliminaries Proof. Let xn n N be a sequence of points in X convergent to a point x X { } 2 2 and let U (f(x)) in Y . It is clear from Definition 1.1.32 and Definition 1.1.5 1 2F that f − (U) (x). Since xn n N converges to x,thereexistsN N s.t. 1 2F { } 2 2 xn f − (U) for all n N.Thenf(xn) U for all n N. Hence, f(xn) n N 2 ≥ 2 ≥ { } 2 converges to f(x). 1.1.5 Hausdor↵spaces Definition 1.1.40. A topological space X is said to be Hausdor↵ (or sepa- rated) if any two distinct points of X have neighbourhoods without common points; or equivalently if: (T2) two distinct points always lie in disjoint open sets. In literature, the Hausdor↵space are often called T2-spaces and the axiom (T2) is said to be the separation axiom. Proposition 1.1.41. In a Hausdor↵space the intersection of all closed neigh- bourhoods of a point contains the point alone. Hence, the singletons are closed. Proof. Let us fix a point x X,whereX is a Hausdor↵space. Denote 2 by C the intersection of all closed neighbourhoods of x. Suppose that there exists y C with y = x. By definition of Hausdor↵space, there exist a 2 6 neighbourhood U(x) of x and a neighbourhood V (y) of y s.t. U(x) V (y)= . \ ; Therefore, y/U(x) because otherwise any neighbourhood of y (in particular 2 V (y)) should have non-empty intersection with U(x). -
The Topology of Subsets of Rn
Lecture 1 The topology of subsets of Rn The basic material of this lecture should be familiar to you from Advanced Calculus courses, but we shall revise it in detail to ensure that you are comfortable with its main notions (the notions of open set and continuous map) and know how to work with them. 1.1. Continuous maps “Topology is the mathematics of continuity” Let R be the set of real numbers. A function f : R −→ R is called continuous at the point x0 ∈ R if for any ε> 0 there exists a δ> 0 such that the inequality |f(x0) − f(x)| < ε holds for all x ∈ R whenever |x0 − x| <δ. The function f is called contin- uous if it is continuous at all points x ∈ R. This is basic one-variable calculus. n Let R be n-dimensional space. By Or(p) denote the open ball of radius r> 0 and center p ∈ Rn, i.e., the set n Or(p) := {q ∈ R : d(p, q) < r}, where d is the distance in Rn. A function f : Rn −→ R is called contin- n uous at the point p0 ∈ R if for any ε> 0 there exists a δ> 0 such that f(p) ∈ Oε(f(p0)) for all p ∈ Oδ(p0). The function f is called continuous if it is continuous at all points p ∈ Rn. This is (more advanced) calculus in several variables. A set G ⊂ Rn is called open in Rn if for any point g ∈ G there exists a n δ> 0 such that Oδ(g) ⊂ G. -
The Non-Hausdorff Number of a Topological Space
THE NON-HAUSDORFF NUMBER OF A TOPOLOGICAL SPACE IVAN S. GOTCHEV Abstract. We call a non-empty subset A of a topological space X finitely non-Hausdorff if for every non-empty finite subset F of A and every family fUx : x 2 F g of open neighborhoods Ux of x 2 F , \fUx : x 2 F g 6= ; and we define the non-Hausdorff number nh(X) of X as follows: nh(X) := 1 + supfjAj : A is a finitely non-Hausdorff subset of Xg. Using this new cardinal function we show that the following three inequalities are true for every topological space X d(X) (i) jXj ≤ 22 · nh(X); 2d(X) (ii) w(X) ≤ 2(2 ·nh(X)); (iii) jXj ≤ d(X)χ(X) · nh(X) and (iv) jXj ≤ nh(X)χ(X)L(X) is true for every T1-topological space X, where d(X) is the density, w(X) is the weight, χ(X) is the character and L(X) is the Lindel¨ofdegree of the space X. The first three inequalities extend to the class of all topological spaces d(X) Pospiˇsil'sinequalities that for every Hausdorff space X, jXj ≤ 22 , 2d(X) w(X) ≤ 22 and jXj ≤ d(X)χ(X). The fourth inequality generalizes 0 to the class of all T1-spaces Arhangel ski˘ı’s inequality that for every Hausdorff space X, jXj ≤ 2χ(X)L(X). It is still an open question if 0 Arhangel ski˘ı’sinequality is true for all T1-spaces. It follows from (iv) that the answer of this question is in the afirmative for all T1-spaces with nh(X) not greater than the cardianilty of the continuum. -
Foliations of Three Dimensional Manifolds∗
Foliations of Three Dimensional Manifolds∗ M. H. Vartanian December 17, 2007 Abstract The theory of foliations began with a question by H. Hopf in the 1930's: \Does there exist on S3 a completely integrable vector field?” By Frobenius' Theorem, this is equivalent to: \Does there exist on S3 a codimension one foliation?" A decade later, G. Reeb answered affirmatively by exhibiting a C1 foliation of S3 consisting of a single compact leaf homeomorphic to the 2-torus with the other leaves homeomorphic to R2 accumulating asymptotically on the compact leaf. Reeb's work raised the basic question: \Does every C1 codimension one foliation of S3 have a compact leaf?" This was answered by S. Novikov with a much stronger statement, one of the deepest results of foliation theory: Every C2 codimension one foliation of a compact 3-dimensional manifold with finite fundamental group has a compact leaf. The basic ideas leading to Novikov's Theorem are surveyed here.1 1 Introduction Intuitively, a foliation is a partition of a manifold M into submanifolds A of the same dimension that stack up locally like the pages of a book. Perhaps the simplest 3 nontrivial example is the foliation of R nfOg by concentric spheres induced by the 3 submersion R nfOg 3 x 7! jjxjj 2 R. Abstracting the essential aspects of this example motivates the following general ∗Survey paper written for S.N. Simic's Math 213, \Advanced Differential Geometry", San Jos´eState University, Fall 2007. 1We follow here the general outline presented in [Ca]. For a more recent and much broader survey of foliation theory, see [To]. -
TOPOLOGY and ITS APPLICATIONS the Number of Complements in The
TOPOLOGY AND ITS APPLICATIONS ELSEVIER Topology and its Applications 55 (1994) 101-125 The number of complements in the lattice of topologies on a fixed set Stephen Watson Department of Mathematics, York Uniuersity, 4700 Keele Street, North York, Ont., Canada M3J IP3 (Received 3 May 1989) (Revised 14 November 1989 and 2 June 1992) Abstract In 1936, Birkhoff ordered the family of all topologies on a set by inclusion and obtained a lattice with 1 and 0. The study of this lattice ought to be a basic pursuit both in combinatorial set theory and in general topology. In this paper, we study the nature of complementation in this lattice. We say that topologies 7 and (T are complementary if and only if 7 A c = 0 and 7 V (T = 1. For simplicity, we call any topology other than the discrete and the indiscrete a proper topology. Hartmanis showed in 1958 that any proper topology on a finite set of size at least 3 has at least two complements. Gaifman showed in 1961 that any proper topology on a countable set has at least two complements. In 1965, Steiner showed that any topology has a complement. The question of the number of distinct complements a topology on a set must possess was first raised by Berri in 1964 who asked if every proper topology on an infinite set must have at least two complements. In 1969, Schnare showed that any proper topology on a set of infinite cardinality K has at least K distinct complements and at most 2” many distinct complements. -
Data Monoids∗
Data Monoids∗ Mikolaj Bojańczyk1 1 University of Warsaw Abstract We develop an algebraic theory for languages of data words. We prove that, under certain conditions, a language of data words is definable in first-order logic if and only if its syntactic monoid is aperiodic. 1998 ACM Subject Classification F.4.3 Formal Languages Keywords and phrases Monoid, Data Words, Nominal Set, First-Order Logic Digital Object Identifier 10.4230/LIPIcs.STACS.2011.105 1 Introduction This paper is an attempt to combine two fields. The first field is the algebraic theory of regular languages. In this theory, a regular lan- guage is represented by its syntactic monoid, which is a finite monoid. It turns out that many important properties of the language are reflected in the structure of its syntactic monoid. One particularly beautiful result is the Schützenberger-McNaughton-Papert theorem, which describes the expressive power of first-order logic. Let L ⊆ A∗ be a regular language. Then L is definable in first-order logic if and only if its syntactic monoid ML is aperiodic. For instance, the language “words where there exists a position with label a” is defined by the first-order logic formula (this example does not even use the order on positions <, which is also allowed in general) ∃x. a(x). The syntactic monoid of this language is isomorphic to {0, 1} with multiplication, where 0 corresponds to the words that satisfy the formula, and 1 to the words that do not. Clearly, this monoid does not contain any non-trivial group. There are many results similar to theorem above, each one providing a connection between seemingly unrelated concepts of logic and algebra, see e.g. -