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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.
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.
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.
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.
Long line; Real line, R; Real projective line, RP 1 ≅ S 1; 2-manifolds. ... Topological manifold; Manifolds with additional structure. Almost complex manifold;
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:
The most familiar non-Hausdorff manifold is the line with two origins, [1] or bug-eyed line. This is the quotient space of two copies of the real line, R × { a } {\displaystyle \mathbb {R} \times \{a\}} and R × { b } {\displaystyle \mathbb {R} \times \{b\}} (with a ≠ b {\displaystyle a\neq b} ), obtained by identifying points ( x , a ...
The manifold can be constructed by first plumbing together disc bundles of Euler number 2 over the sphere, according to the Dynkin diagram for . This results in P E 8 {\displaystyle P_{E_{8}}} , a 4-manifold whose boundary is homeomorphic to the Poincaré homology sphere .