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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-manifold with boundary. Its boundary is a circle, a 1-manifold.
A boundary point of a set is any element of that set's boundary. The boundary defined above is sometimes called the set's topological boundary to distinguish it from other similarly named notions such as the boundary of a manifold with boundary or the boundary of a manifold with corners, to name just a few examples.
A Seifert surface of a knot is however a manifold with boundary, the boundary being the knot, i.e. homeomorphic to the unit circle. The genus of such a surface is defined to be the genus of the two-manifold, which is obtained by gluing the unit disk along the boundary.
In mathematics, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fiber bundles.
Definition. An open book decomposition of a 3-dimensional manifold M is a pair (B, π) where . B is an oriented link in M, called the binding of the open book;; π: M \ B → S 1 is a fibration of the complement of B such that for each θ ∈ S 1, π −1 (θ) is the interior of a compact surface Σ ⊂ M whose boundary is B.
The boundary of a manifold is a manifold , which has dimension . An orientation on M {\displaystyle M} induces an orientation on ∂ M {\displaystyle \partial M} . We usually denote a submanifold by Σ ⊂ M {\displaystyle \Sigma \subset M} .
Andrew Casson introduces the Casson invariant for homology 3-spheres, bringing the whole new set of ideas into the 3-dimensional topology, and relating the geometry of 3-manifolds with the geometry of representation spaces of the fundamental group of a 2-manifold. This leads to a direct connection with mathematical physics.
A manifold is metrizable if and only if it is paracompact. The long line is an example a normal Hausdorff 1-dimensional topological manifold that is not metrizable nor paracompact. Since metrizability is such a desirable property for a topological space, it is common to add paracompactness to the definition of a manifold.