7/3/2023 0 Comments Homeomorphisms of hyperspaces![]() It is thus important to realize that it is the formal definition given above that counts. The intuitive criterion of stretching, bending, cutting and gluing back together takes a certain amount of practice to apply correctly-it may not be obvious from the description above that deforming a line segment to a point is impermissible, for instance. Some homeomorphisms are not continuous deformations, such as the homeomorphism between a trefoil knot and a circle.Īn often-repeated mathematical joke is that topologists cannot tell the difference between a coffee cup and a donut, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while preserving the donut hole in the cup's handle.Ī function f : X → Y ( Alexander's trick). Some continuous deformations are not homeomorphisms, such as the deformation of a line into a point. However, this description can be misleading. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. Very roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. Homeomorphisms are the isomorphisms in the category of topological spaces-that is, they are the mappings that preserve all the topological properties of a given space. In the mathematical field of topology, a homeomorphism (from Greek ὅμοιος (homoios) 'similar, same', and μορφή (morphē) 'shape, form', named by Henri Poincaré ), topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. But there need not be a continuous deformation for two spaces to be homeomorphic – only a continuous mapping with a continuous inverse function. Furthermore, in the continuous complete case, the d_H-Scott topology coincides with the lower Vietoris topology, and the d_Q-Scott topology coincides with the upper Vietoris topology.A continuous deformation between a coffee mug and a donut ( torus) illustrating that they are homeomorphic. Then we show that the Hoare and Smyth powerdomains of an algebraic complete quasi-metric space are again algebraic complete, with those quasi-metrics, and similarly that the corresponding powerdomains of continuous complete quasi-metric spaces are continuous complete. Through these isomorphisms again, the two powerdomains inherit quasi-metrics d_H and d_Q, respectively, that are reminiscent of the well-known Hausdorff metric. Turning to the corresponding hyperspaces, namely the same powerdomains, but equipped with the lower Vietoris and upper Vietoris topologies instead, this turns into homeomorphisms with the corresponding space of previsions, equipped with the so-called weak topology. ![]() There are natural isomorphisms between the Hoare and Smyth powerdomains, as used in denotational semantics, and spaces of discrete sublinear previsions, and of discrete normalized superlinear previsions, respectively. We show that the Kantorovich-Rubinstein quasi-metrics d_KR and d^a_KR of Part I extend naturally to various spaces of previsions, and in particular not just the linear previsions (roughly, measures) of Part I. ![]()
0 Comments
Leave a Reply. |