Bagpipe Theorem

picture info	Bagpipe Theorem In mathematics, the bagpipe theorem of describes the structure of the connected (but possibly non-paracompact) ω-bounded surfaces by showing that they are "bagpipes": the connected sum of a compact "bag" with several "long pipes". Statement A space is called ω-bounded if the closure of every countable set is compact. For example, the long line and the closed long ray are ω-bounded but not compact. When restricted to a metric space ω-boundedness is equivalent to compactness. The bagpipe theorem states that every ω-bounded connected surface is the connected sum of a compact connected surface and a finite number of long pipes. A space P is called a long pipe if there exist subspaces \ each of which is homeomorphic to S^1 \times \mathbb such that for n we have $\overline \subseteq U_m$ and the boundary of $U_n$ in $U_m$ is homeomorphic to $S^1$ . The simplest example of a pipe is the product $S^1 \times L^+$ [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Chambers 1908 Bagpipe Chambers may refer to: Places Canada: Chambers Township, Ontario United States: Chambers County, Alabama Chambers, Arizona, an unincorporated community in Apache County Chambers, Nebraska * Chambers, West Virginia Chambers Township, Holt County, Nebraska Chambers Branch, a stream in Kansas Chambers County, Texas Other ''Chambers Dictionary'' of the English Language * Chambers Harrap, the publishers of Chambers Dictionary * Chambers and Partners, a British organisation that produces international rankings for the legal industry * Chambers of parliament * ''Chambers'' (album), by Steady & Co. (2001) * Hedingham & Chambers, a bus company in Suffolk and Essex * judge's chambers, a judge's office where some matters are heard out of court * barristers' chambers, in some English-speaking countries a set of rooms from which barristers practice * ''Chambers'' (series), a BBC Radio 4 legal sitcom starring John Bird which later moved to television * Chambers stove, a defunct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Paracompact Space In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it admits partitions of unity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff. Every closed subspace of a paracompact space is paracompact. While compact subsets of Hausdorff spaces are always closed, this is not true for paracompact subsets. A space such that every subspace of it is a paracompact space is called hereditarily paracompact. This is equivalent to requiring that every open subspace be paracompact. Tychonoff's theorem (which states that the product of any collection of compact topological spaces is compact) does not generalize to paracompact spaces in that the product of paracompact spaces need not be paracompact. Howeve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	ω-bounded Space In mathematics, an ω-bounded space is a topological space in which the closure of every countable subset is compact. More generally, if ''P'' is some property of subspaces, then a ''P''-bounded space is one in which every subspace with property ''P'' has compact closure. Every compact space is ω-bounded, and every ω-bounded space is countably compact. The long line is ω-bounded but not compact. The bagpipe theorem In mathematics, the bagpipe theorem of describes the structure of the connected (but possibly non-paracompact space, paracompact) ω-bounded space, ω-bounded surfaces by showing that they are "bagpipes": the connected sum of a compact space, co ... describes the ω-bounded surfaces. References * Properties of topological spaces {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Connected Sum In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classification of closed surfaces. More generally, one can also join manifolds together along identical submanifolds; this generalization is often called the fiber sum. There is also a closely related notion of a connected sum on knots, called the knot sum or composition of knots. Connected sum at a point A connected sum of two ''m''-dimensional manifolds is a manifold formed by deleting a ball inside each manifold and gluing together the resulting boundary spheres. If both manifolds are oriented, there is a unique connected sum defined by having the gluing map reverse orientation. Although the construction uses the choice of the balls, the result is unique up to homeomorphism. One can also make this operation work in the smooth categor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Compact Space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topologic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]