If M is a closed, oriented manifold and if M'is obtained from M by removing an open ball, then the connected sum M \ mathrel { \ # }-M is the double of M '.
32.
Let U ( n ) be the set of all open balls of radius 1 / n; clearly, U ( n ) is a cover of Y, so a finite subcover F ( n ) exists.
33.
The regularity assumption on \ Omega can be replaced with an interior ball condition : the lemma holds provided that there exists an open ball B \ subset \ Omega with x _ 0 \ in \ partial B.
34.
This V ( x _ 1, x _ 2 ) is simply the scaled energy of the system Clearly, V ( x _ 1, x _ 2 ) is positive definite in an open ball of radius \ pi around the origin.
35.
Looking like a giant gyroscope, tilted as if it were about to spin off into space, the open ball of interlocking aluminum rings and stainless steel fittings is called an armillary, an astronomical instrument that has been used since antiquity to depict the universe.
36.
Each choice of open sets for a space is called a varieties, and the topology on a differential manifold in differential topology where each point within the space is contained in an open set that is homeomorphic to an open ball in a finite-dimensional Euclidean space.
37.
If " S " is a subset of a Euclidean space, then " x " is an interior point of " S " if there exists an open ball centered at " x " which is completely contained in " S " . ( This is illustrated in the introductory section to this article .)
38.
A topological space is a set " X " together with a topology on " X ", which is a set of subsets of " X " satisfying a few requirements with respect to their unions and intersections that generalize the properties of the open balls in metric spaces while still allowing to talk about the neighbourhoods of a given point.
39.
I know in the case of the general metric space ( X, d ) with an open ball A it is not the case that \ { y : d ( x, y ) = r \ } \ subset B ( A ) but I cannot see where my proof below explicitly assumes that the metric space is R ^ n.
40.
The idea of the topological space was invented in the late 19th and early 20th century as an attempt to generalize the idea of the metric space, and the'open set'was a generalization of the'open ball'of a metric space; metric spaces in turn were a generalization of Euclidean spaces, invented in an attempt to get better insight into problems of real and complex analysis.
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