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As of November 1988 MIT had 722 workstations in 33 private and public clusters on and off campus, including student living groups and fraternities. A survey found that 92% of undergraduates had used the Athena workstations at least once, and 25% used them every day. [5] [9] The project received an extension of three years in January 1988.
A map with twelve pentagonal faces. In topology and graph theory, a map is a subdivision of a surface such as the Euclidean plane into interior-disjoint regions, formed by embedding a graph onto the surface and forming connected components (faces) of the complement of the graph.
The following sets will constitute the basic open subsets of topologies on spaces of linear maps. For any subsets and , let (,):= {: ()}.. The family {(,):,} forms a neighborhood basis [1] at the origin for a unique translation-invariant topology on , where this topology is not necessarily a vector topology (that is, it might not make into a TVS).
Orientation-preserving maps are precisely those that act trivially on top cohomology H 2 (Σ) ≅ Z. H 1 (Σ) has a symplectic structure, coming from the cup product ; since these maps are automorphisms, and maps preserve the cup product, the mapping class group acts as symplectic automorphisms, and indeed all symplectic automorphisms are ...
Topological geometry deals with incidence structures consisting of a point set and a family of subsets of called lines or circles etc. such that both and carry a topology and all geometric operations like joining points by a line or intersecting lines are continuous.
The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology. N.B. Some authors use the name Top for the categories with topological manifolds , with compactly generated spaces as objects and continuous maps as morphisms or with the category of compactly generated weak ...
In mathematics, pointless topology, also called point-free topology (or pointfree topology) and locale theory, is an approach to topology that avoids mentioning points, and in which the lattices of open sets are the primitive notions. [1] In this approach it becomes possible to construct topologically interesting spaces from purely algebraic ...
A basic example in topology is lifting a path in one topological space to a path in a covering space. [1] For example, consider mapping opposite points on a sphere to the same point, a continuous map from the sphere covering the projective plane. A path in the projective plane is a continuous map from the unit interval [0,1]. We can lift such a ...