Topological Properties of the Ordinal Spaces SΩ and SΩ Topology II
... (1) SΩ is not connected: Let x be an element of SΩ and let x1 be its immediate successor.
Then [a0 , x1 ) and (x, +∞) are disjoint open sets and they are nonempty because x ∈
[a0 , x1 ) and x1 ∈ (x, +∞). So these two sets form a separation of SΩ , showing that SΩ
is not connected. (Note that neither ...
Fundamental group
In the mathematics of algebraic topology, the fundamental group is a mathematical group associated to any given pointed topological space that provides a way to determine when two paths, starting and ending at a fixed base point, can be continuously deformed into each other. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a topological invariant: homeomorphic topological spaces have the same fundamental group.Fundamental groups can be studied using the theory of covering spaces, since a fundamental group coincides with the group of deck transformations of the associated universal covering space. The abelianization of the fundamental group can be identified with the first homology group of the space. When the topological space is homeomorphic to a simplicial complex, its fundamental group can be described explicitly in terms of generators and relations.Henri Poincaré defined the fundamental group in 1895 in his paper ""Analysis situs"". The concept emerged in the theory of Riemann surfaces, in the work of Bernhard Riemann, Poincaré, and Felix Klein. It describes the monodromy properties of complex-valued functions, as well as providing a complete topological classification of closed surfaces.
