- #1
lugita15
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- 15
Let ##d_1## and ##d_2## be two metrics on the same set ##X##. Suppose that a set is open with respect to ##d_1## if and only if it is open with respect to ##d_2##, and a set is bounded with respect to ##d_1## it and only if it is bounded with respect to ##d_2##. (In technical language, ##d_1## and ##d_2## induce the same topology and the same bornology.) My question is, does this imply that a sequence is Cauchy with respect to ##d_1## if and only it is Cauchy with respect to ##d_2##?
If not, does anyone know of an example of two metrics which share the same open sets and the same bounded sets, but have different collections of Cauchy sequences?
If not, does anyone know of an example of two metrics which share the same open sets and the same bounded sets, but have different collections of Cauchy sequences?