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• ID : Q202906

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1. Homeomorphism, in mathematics, a correspondence between two figures or surfaces or other geometrical objects, defined by a one-to-one mapping that is continuous in both directions.
2. A topological property is defined to be a property that is preserved under a homeomorphism.
3. Two spaces are called topologically equivalent if there exists a homeomorphism between them.
4. Any simple polygon is homeomorphic to a circle; all figures homeomorphic to a circle are called simple closed curves.
5. Eventually Kuratowski showed in 1921 that in the plane there can be a continuous bijection of P onto Q, and of Q onto P, without P and Q being homeomorphic.
6. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same.
7. 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.
8. A homeomorphism is sometimes called a bicontinuous function.
9. If such a function exists, X {\displaystyle X} and Y {\displaystyle Y} are homeomorphic.
10. → Y that is continuous, and whose inverse f−1 is also continuous, with respect to the given topologies; such a function f is called a homeomorphism.
11. X → Y is a homeomorphism if and only if both f and f−1 map open sets to open sets.
12. It is therefore useful to be able to determine whether two given topological spaces are homeomorphic.
13. Of course, if we can find a specific homeomorphism between them then the question is answered; but if we fail to find a homeomorphism then we cannot deduce that there isn't one.
14. A homeomorphism which also preserves distances is called an isometry.
15. Even so, the homeomorphism problem remains highly important.
16. In the topology of manifolds it was only in the late 1960s that methods for studying manifolds up to a homeomorphism were developed.
17. Thus, a compact Hausdorff space is homeomorphic to the space of maximal ideals of the algebra of real functions defined on it.
18. If two spaces are homeomorphic, then the method of spectra (and of diminishing subdivisions) is the only one of general value for the establishment of homeomorphism.
19. In general topology, a homeomorphism is a map between spaces that preserves all topological properties.
20. In topology, the most basic equivalence is a homeomorphism, which allows spaces that appear quite different in most other subjects to be declared equivalent in topology.
21. Two topological spaces and are said to be homeomorphic, denoted by , if there exists a homeomorphism between them.
22. This kind of homeomorphism can be generalized substantially using linear algebra.
23. If a subset,, can be mapped to another,, via a nonsingular linear transformation, thenandare homeomorphic.
24. What two sets of points does this homeomorphism show to be topologically equivalent?
25. Describe a stereographic projection projection and justify why it is a homeomorphism.
26. I have no texts that describe homeomorphism or stereographic projection.
27. The function f is called a homeomorphism.
28. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids), algebraically in terms of the surface braid group.
29. So the reader might think that he should read the works of Poincaré and Cantor if he wished to find the origin of the concept of homeomorphism.
30. Since Poincaré’s 1895 article is the origin of the word “homeomorphism” but not of the concept of homeomorphism, how did that concept originate?
31. At the time that Poincaré wrote, there was a second and broader concept of homeomorphism in use.
32. Thus Seifert and Threlfall accepted the modern definition of homeomorphism.
33. Section 4 presents an algorithm to get an homeomorphism between two models and outlines two morphing processes using FDTs.
34. Two models are said homeomorphic if there exists an homeomorphism.
35. Whenever is Tychonoff, since any self-homeomorphism of continuously extends to , the Stone- ech compactification of , then embeds as a subgroup in .
36. We say that a -compactification of has the lifting property if every self-homeomorphism of continuously extends to .
37. Whenever is a -compactification of with the lifting property, the homeomorphism group embeds as subgroup in equipped with the compact-open topology.
38. This issue is essentially achieved by the property: any two non-empty clopen subspaces of are homeomorphic, as it is derived from the topological characterization of .

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위키데이터

• ID : Q202906