"A manifold, roughly, is a topological space in which some neighborhood of each point admits a coordinate system, consisting of real coordinate functions on the points of the neighborhood, which determine the position of points and the topology of that neighborhood; that is, the space is locally cartesian. Moreover, the passage from one coordinate system to another is smooth in the overlapping region, so that the meaning of 'differentiable' curve, function, or map is consistent when referred to either system." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)
"A manifold can be given by specifying the coordinate ranges
of an atlas, the images in those coordinate ranges of the overlapping parts of
the coordinate domains, and the coordinate transformations for each of those
overlapping domains. When a manifold is specified in this way, a rather tricky
condition on the specifications is needed to give the Hausdorff property, but
otherwise the topology can be defined completely by simply requiring the
coordinate maps to be homeomorphisms."
"A set is countable if it is either finite or its members can
be arranged in an infinite sequence; or, what is the same, there is a 1-1 map
from the set into the positive integers."
"An initial study of tensor analysis can. almost ignore the
topological aspects since the topological assumptions are either very natural
(continuity, the Hausdorff property) or highly technical (separability,
paracompactness). However, a deeper analysis of many of the existence problems
encountered in tensor analysis requires assumption of some of the more
difficult-to-use topological properties, such as compactness and paracompactness."
"In the definition of a coordinate system we have required that the coordinate neighborhood and the range in Rd be open sets. This is contrary to popular usage, or at least more specific than the usage of curvilinear coordinates in advanced calculus. For example, spherical coordinates are used even along points of the z axis where they are not even 1-1. The reasons for the restriction to open sets are that it forces a uniformity in the local structure which simplifies analysis on a manifold (there are no 'edge points') and, even if local uniformity were forced in some other way, it avoids the problem of. spelling out what we mean by differentiability at boundary points of the coordinate neighborhood; that is, one-sided derivatives need not be mentioned. On the other hand, in applications, boundary value problems frequently arise, the setting for which is a manifold with boundary. These spaces are more general than manifolds and the extra generality arises from allowing a boundary manifold of one dimension less. The points of the boundary manifold have a coordinate neighborhood in the boundary manifold which is attached to a coordinate neighborhood of the interior in much the same way as a face of a cube is attached to the interior. Just as the study of boundary value problems is more difficult than the study of spatial problems, the study of manifolds with boundary is more difficult than that of mere manifolds, so we shall limit ourselves to the latter." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)
"Set theory is concerned with abstract objects and their
relation to various collections which contain them. We do not define what a set
is but accept it as a primitive notion. We gain an intuitive feeling for the
meaning of sets and, consequently, an idea of their usage from merely listing
some of the synonyms: class, collection, conglomeration, bunch, aggregate.
Similarly, the notion of an object is primitive, with synonyms element and
point. Finally, the relation between elements and sets, the idea of an element
being in a set, is primitive."
"The mathematical models for many physical systems have manifolds as the basic objects of study, upon which further structure may be defined to obtain whatever system is in question. The concept generalizes and includes the special cases of the cartesian line, plane, space, and the surfaces which are studied in advanced calculus. The theory of these spaces which generalizes to manifolds includes the ideas of differentiable functions, smooth curves, tangent vectors, and vector fields. However, the notions of distance between points and straight lines (or shortest paths) are not part of the idea of a manifold but arise as consequences of additional structure, which may or may not be assumed and in any case is not unique." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)
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