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It is common to place additional requirements on topological manifolds. In particular, many authors define them to be paracompact [3] or second-countable. [2] In the remainder of this article a manifold will mean a topological manifold. An n-manifold will mean a topological manifold such that every point has a neighborhood homeomorphic to R n.
By definition, all manifolds are topological manifolds, so the phrase "topological manifold" is usually used to emphasize that a manifold lacks additional structure, or that only its topological properties are being considered. Formally, a topological manifold is a topological space locally homeomorphic to a Euclidean space.
This is a list of particular manifolds, by Wikipedia page. See also list of geometric topology topics . For categorical listings see Category:Manifolds and its subcategories.
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.
A topological manifold that is in the image of is said to "admit a differentiable structure", and the fiber over a given topological manifold is "the different differentiable structures on the given topological manifold". Thus given two categories, the two natural questions are:
In geometry, a tame manifold is a manifold with a well-behaved compactification. More precisely, a manifold M {\displaystyle M} is called tame if it is homeomorphic to a compact manifold with a closed subset of the boundary removed.
Conversely, the boundary of a closed disk viewed as a manifold is the bounding circle, as is its topological boundary viewed as a subset of the real plane, while its topological boundary viewed as a subset of itself is empty. In particular, the topological boundary depends on the ambient space, while the boundary of a manifold is invariant.
More generally, a (topological) surface with boundary is a Hausdorff topological space in which every point has an open neighbourhood homeomorphic to some open subset of the closure of the upper half-plane H 2 in C. These homeomorphisms are also known as (coordinate) charts. The boundary of the upper half-plane is the x-axis.