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:If we have a space where Cauchy sequences are meaningful (a metric space, i.e. a space where distance is defined), a standard procedure to force all Cauchy sequences to converge is adding new points to the space (a process called completing?). When applied to the rational numbers, it gives the following useful [construction of the real numbers]?: It should point out that this only works with a Euclidean metric or one equivalent to a Euclidean metric; using other metrics gives you the p-adic numbers instead. |
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:If we have a space where Cauchy sequences are meaningful (a :metric space, i.e. a space where distance is defined), a :standard procedure to force all Cauchy sequences to converge is :adding new points to the space (a process called completing?). :When applied to the rational numbers, it gives the following :useful [construction of the real numbers]?: It should point out that this only works with a Euclidean metric or one equivalent to a Euclidean metric; using other metrics gives you the p-adic numbers instead. The following comment was moved from the main page: RB: The dimension is actually difficult to define: the reals have dimension 1 for pretty much any sensible definition, but the best definition I know is that cohomology with compact support is non trivial in dimension 1 and vanishes above it. |
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