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  1. Copilot Answer
    Motivating examples

    After a line, a circle is the simplest example of a topological manifold. Topology ignores bending, so a small piece of a circle is treated the same as a small piece of a line. Considering, for instance, the top part of the unit circle, … See more

    Mathematical definition

    Informally, a manifold is a space that is "modeled on" Euclidean space.
    There are many different kinds of manifolds. In geometry and topology, all manifolds are topological manifolds, possibly wit… See more

    Charts, atlases, and transition maps

    The spherical Earth is navigated using flat maps or charts, collected in an atlas. Similarly, a manifold can be described using mathematical maps, called coordinate charts, collected in a mathematical atlas. It is not gene… See more

    Manifold with boundary

    A manifold with boundary is a manifold with an edge. For example, a sheet of paper is a 2-manifold with a 1-dimensional boundary. The boundary of an -manifold with boundary is an -manifold. A disk (circle plus interior) is a 2-manifol… See more

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