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How would one go about proving ℝn is not homeomorphic to ℝm for m≠n. This isn't a homework question, I was told, but we weren't shown the proof.Is there somewhere with a proof I can see, or can someone outline it briefly?
6.28318531 said:How would one go about proving ℝn is not homeomorphic to ℝm for m≠n. This isn't a homework question, I was told, but we weren't shown the proof.Is there somewhere with a proof I can see, or can someone outline it briefly?
mathwonk said:Lavinia, I agree your idea to deduce the continuous case from the differentiable case by approximation, can likely be made to work. nice!
In fact, Milnor's beautiful book Topology from the differentiable viewpoint contains such an argument for Brouwer's fix point theorem. I.e. if there were a continuous self map of the disc with no fix point, then every point would move at least e>0. If we approximate this map within e/2 by a smooth map, then every point still moves a positive amount. Thus the fact every smooth self map has a fix point implies that every continuous one does too.
A homeomorphism is a type of function between two topological spaces that preserves the basic structure of those spaces. In simpler terms, it is a continuous and bijective mapping between two spaces that allows for stretching, twisting, and bending of the space, but does not allow for tearing or gluing of points.
A homeomorphism and an isomorphism are both types of mappings between spaces, but they differ in the type of structure they preserve. While a homeomorphism preserves the basic topological structure, an isomorphism preserves algebraic structure, such as addition or multiplication.
This is because the dimensions of R^n and R^m are different, meaning they have different underlying structures. A homeomorphism requires a bijective mapping between two spaces, which is impossible if the number of dimensions is not the same for both spaces.
Yes, as long as n and m are equal, R^n and R^m can be homeomorphic. This is because the number of dimensions is the same, so a bijective mapping between the spaces is possible.
Yes, homeomorphisms have various applications in fields such as physics, engineering, and computer science. For example, they can be used in simulations to model the behavior of physical systems, in image recognition and compression algorithms, and in designing efficient networks and circuits.