A projective limit ( or a filtered limit ) of rings is defined as follows.
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More generally, every locally compact, almost connected group is the projective limit of a Lie group.
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An important application of an infinite direct product is the construction of a projective limit of rings ( see below ).
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By taking the projective limit of the pro-representable functor in the larger category of linearly topologized local rings, one obtains a complete linearly topologized local ring representing the functor.
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:If a connected locally compact group is a projective limit of a sequence of Lie groups, and if " has no small subgroups " ( a condition defined below ), then is a Lie group.
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The ring \ mathbb Z _ p of p-adic integers can be understood as the projective limit of \ mathbb { Z } / p ^ i \ mathbb { Z }, taking i \ to \ infty.
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