Topology, Graphs and Complexes скачать в хорошем качестве

Topology, Graphs and Complexes

More reflection on some ongoing work in finite mathematics aiming to avoid any infinity and especially Euclidean spaces. Most extobooks in topology refer to the continuum, even Alexandroff, the first serious pioneer in the use of finite topological spaces. Finite topologies on finite abstract simplicial complexes are ( like all finite topological spaces) Alexandroff : every point has a smallest open neighorhood. The topology has the stars as sub-basis: the sets U(x)={y, y contains x} as basis elements. How do we define homeomorphism between two such topological spaces? The standard notion of continuity but homeomorphism would be too narrow as it would just produce a bijection between the topological spaces which in the finite case is nonsense. We want for example an icosahedron homeomorphic to an octahedron. We propose the following definition: H is a continuous image of G, if there is a Barycentric refinement G' of G and a continuous map f:G' to H such that the inverse image of every unit sphere S(x) = boundary of U(x) is homeomorphic to S(x) and that for every locally maximal x, the ball B(x), the closure of U(x) has an inverse image that is a ball in the finite sense. Two spaces are homeomorphic if any of them is a continuous image of the other. The definition is recursive in dimension and so constructive. If one space is a manifold then any space homeomorphic is also a manifold. The property of d-manifold is also inductively defines as a complex in which every unit sphere is a (d-1) sphere. A d-sphere is a d-manifold such that G-U(x) is contractible for some x. And a space is contractible (again inductively defined) if there exists a point such that S(x) and G-U(x) are both contractible. Note that since U(x) is open, G-U(x) is closed and so a simplicial complex. P.S. The drone footage at the beginning and end was captured earlier in the week in the Fells reservation. Saturday, the recording in my office was again a rainy day. Update January 8 2023, here is something written down about topology: https://people.math.harvard.edu/~knil... Will be mostly a seed for some other things.