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The Euler characteristic can be defined for connected plane graphs by the same + formula as for polyhedral surfaces, where F is the number of faces in the graph, including the exterior face. The Euler characteristic of any plane connected graph G is 2.
It has Euler characteristic 1, hence a demigenus (non-orientable genus, Euler genus) of 1. The topological real projective plane can be constructed by taking the (single) edge of a Möbius strip and gluing it to itself in the correct direction, or by gluing the edge to a disk. Alternately, the real projective plane can be constructed by ...
Euler's formula states that if a finite, connected, planar graph is drawn in the plane without any edge intersections, and v is the number of vertices, e is the number of edges and f is the number of faces (regions bounded by edges, including the outer, infinitely large region), then
Formally, a complex projective space is the space of complex lines through the origin of an (n+1)-dimensional complex vector space. The space is denoted variously as P(C n+1), P n (C) or CP n. When n = 1, the complex projective space CP 1 is the Riemann sphere, and when n = 2, CP 2 is the complex projective plane (see there for a more ...
The Euler characteristic of the real projective plane is 1, and in general the Euler characteristic of the connected sum of k of them is 2 − k. It follows that a closed surface is determined, up to homeomorphism, by two pieces of information: its Euler characteristic, and whether it is orientable or not.
Alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ, via the relationship χ = 2 − k, where k is the non-orientable genus. For instance: A real projective plane has a non-orientable genus 1. A Klein bottle has non-orientable genus 2.
Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces.
The projective plane over K, denoted PG(2, K) or KP 2, has a set of points consisting of all the 1-dimensional subspaces in K 3. A subset L of the points of PG(2, K) is a line in PG(2, K) if there exists a 2-dimensional subspace of K 3 whose set of 1-dimensional subspaces is exactly L.