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In other words, is fiber-preserving, and f is the induced map on the space of fibers of E: since π E is surjective, f is uniquely determined by . For a given f , such a bundle map φ {\displaystyle \varphi } is said to be a bundle map covering f .
Another example of a pullback comes from the theory of fiber bundles: given a bundle map π : E → B and a continuous map f : X → B, the pullback (formed in the category of topological spaces with continuous maps) X × B E is a fiber bundle over X called the pullback bundle. The associated commutative diagram is a morphism of fiber bundles.
Fiber bundles became their own object of study in the period 1935–1940. The first general definition appeared in the works of Whitney. [11] Whitney came to the general definition of a fiber bundle from his study of a more particular notion of a sphere bundle, [12] that is a fiber bundle whose fiber is a sphere of arbitrary dimension. [13]
Cognitive maps have been studied in various fields, such as psychology, education, archaeology, planning, geography, cartography, architecture, landscape architecture, urban planning, management and history. [6] Because of the broad use and study of cognitive maps, it has become a colloquialism for almost any mental representation or model. [6]
In mathematics, a pullback bundle or induced bundle [1] [2] [3] is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle π : E → B and a continuous map f : B′ → B one can define a "pullback" of E by f as a bundle f * E over B′. The fiber of f * E over a point b′ in B′ is just the fiber of E over f(b′).
Diagram originally published by the Fiber Optics LAN Section of the Telecommunications Industry Association. Fiber to the Edge (FTTE), fiber to the telecom enclosure (FTTTE) or fiber to the zone (FTTZ), [1] is a fiber to the x networking approach used in the enterprise building (hotels, convention centers, office buildings, hospitals, senior living communities, Multi-Dwelling Units, stadiums ...
A principal -bundle, where denotes any topological group, is a fiber bundle: together with a continuous right action such that preserves the fibers of (i.e. if then for all ) and acts freely and transitively (meaning each fiber is a G-torsor) on them in such a way that for each and , the map sending to is a homeomorphism.
A function between topological spaces is called monotone if every fiber is a connected subspace of its domain. A function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is monotone in this topological sense if and only if it is non-increasing or non-decreasing , which is the usual meaning of " monotone function " in real analysis .