Boundary (topology)

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set S include bd(S), fr(S), and ∂S. Some authors (for example Willard, in General Topology) use the term frontier instead of boundary in an attempt to avoid confusion with the concept of boundary used in algebraic topology and manifold theory. However, frontier sometimes refers to a different set, which is the set of boundary points which are not actually in the set; that is, S \ S.

A connected component of the boundary of S is called a boundary component of S.

If the set consists of discrete points only, then the set has only a boundary and no interior.

Boundary

Boundary (plural: boundaries) may refer to

Political geography

Psychology

Sociology

Mathematics and physics

Other

Boundary (cricket)

In cricket a boundary is the edge or boundary of the playing field, or a scoring shot where the ball is hit to or beyond that point.

Edge of the field

The boundary is the edge of the playing field, or the physical object marking the edge of the field, such as a rope or fence. In low-level matches, a series of plastic cones are often used. Since the early 2000s the boundaries at professional matches are often a series of padded cushions carrying sponsors' logos strung along a rope. If it is moved during play (such as by a fielder sliding into the rope) the boundary is considered to remain at the point where that object first stood.

When the cricket ball is inside the boundary, it is live. When the ball is touching the boundary, grounded beyond the boundary, or being touched by a fielder who is himself either touching the boundary or grounded beyond it, it is dead and the batting side usually scores 4 or 6 runs for hitting the ball over the boundary. Because of this rule, fielders near the boundary attempting to intercept the ball while running or diving, often flick the ball back in to the field of play rather than pick it up directly, because their momentum could carry them beyond the rope while holding the ball. They then return to the field to pick the ball up and throw it back to the bowler.

Topological manifold

In topology, a branch of mathematics, a topological manifold is a topological space (which may also be a separated space) which locally resembles real n-dimensional space in a sense defined below. Topological manifolds form an important class of topological spaces with applications throughout mathematics.

A manifold can mean a topological manifold, or more frequently, a topological manifold together with some additional structure. Differentiable manifolds, for example, are topological manifolds equipped with a differential structure. Every manifold has an underlying topological manifold, obtained simply by forgetting the additional structure. An overview of the manifold concept is given in that article. This article focuses purely on the topological aspects of manifolds.

Formal definition

A topological space X is called locally Euclidean if there is a non-negative integer n such that every point in X has a neighborhood which is homeomorphic to the Euclidean space Eⁿ (or, equivalently, to the real n-space Rⁿ, or to some connected open subset of either of two).

Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, that satisfy a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.

Definition

The utility of the notion of a topology is shown by the fact that there are several equivalent definitions of this structure. Thus one chooses the axiomatisation suited for the application. The most commonly used, and the most elegant, is that in terms of open sets, but the most intuitive is that in terms of neighbourhoods and so we give this first. Note: A variety of other axiomatisations of topological spaces are listed in the Exercises of the book by Vaidyanathaswamy.

Topology (musical ensemble)

Topology is an indie classical quintet from Australia, formed in 1997. A leading Australian new music ensemble, they perform throughout Australia and abroad and have to date released 12 albums, including one with rock/electronica band Full Fathom Five and one with contemporary ensemble Loops. They were formerly the resident ensemble at the University of Western Sydney. The group works with composers including Tim Brady in Canada, Andrew Poppy, Michael Nyman, and Jeremy Peyton Jones in the UK, and Terry Riley, Steve Reich, Philip Glass, Carl Stone, and Paul Dresher in the US, as well as many Australian composers.

In 2009, Topology won the Outstanding Contribution by an Organisation award at the Australasian Performing Right Association (APRA) Classical Music Awards for their work on the 2008 Brisbane Powerhouse Series.

Members

Bernard studied viola at the Queensland Conservatorium (B.Mus 1987) and at Michigan State University (Master of Music 1993) with John Graham and Robert Dan. He studied in summer schools with Kim Kashkashian (Aldeborough), the Alban Berg Quartet and the Kronos Quartet. While in the US, he played with the Arlington Quartet, touring the US and UK. He was a violist in the Queensland Philharmonic Orchestra from 1994-2000, and is now Associate Principal Violist of the Queensland Orchestra, playing solo parts in works such as the sixth Brandenburg Concerto. He has directed several concerts for the Queensland Philharmonic’s Off the Factory Floor chamber series.

Topology (journal)

Topology was a peer-reviewed mathematical journal covering topology and geometry. It was established in 1962 and was published by Elsevier. The last issue of Topology appeared in 2009.

Pricing dispute

On 10 August 2006, after months of unsuccessful negotiations with Elsevier about the price policy of library subscriptions, the entire editorial board of the journal handed in their resignation, effective 31 December 2006. Subsequently, two more issues appeared in 2007 with papers that had been accepted before the resignation of the editors. In early January the former editors instructed Elsevier to remove their names from the website of the journal, but Elsevier refused to comply, justifying their decision by saying that the editorial board should remain on the journal until all of the papers accepted during its tenure had been published.

In 2007 the former editors of Topology announced the launch of the Journal of Topology, published by Oxford University Press on behalf of the London Mathematical Society at a significantly lower price. Its first issue appeared in January 2008.