This example involves the dense subspace, and a second that maps this dense set into a countable unit simplex ( an infinite set of points all at unit distance from each other ).
32.
Dually, just as the complement of a nowhere dense set need not be open, but has a dense open sets ), but contains a dense G ? set formed from dense open sets.
33.
As explained above, the points of the plane ( other than the origin ) can be divided into two dense sets which we may call " A " and " B ".
34.
CMJ described the album as a " raw, alien, dense set of four tuneless almost-not-even-songs made of spattered guitar, drums and Tony Pinotti's wrecked, howling voice ".
35.
In mathematics, the "'Fatou conjecture "', named after Pierre Fatou, states that a quadratic family of maps from the complex plane to itself is hyperbolic for an open dense set of parameters.
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The iterations converge to the fixed point of the IFS . Whenever x 0 belongs to the attractor of the IFS, all iterations x k stay inside the attractor and, with probability 1, form a dense set in the latter.
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Conversely, given a dense set D, Zorn s Lemma shows that there exists a maximal antichain A \ subseteq D, and then G \ cap D \ neq \ varnothing if and only if G \ cap A \ neq \ varnothing.
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This can be done either with a phased array where the coils are combined in parallel and often sample overlapping areas with gaps in the sampling or with massive coil arrays, which are a much denser set of receivers separate from the excitation coils.
39.
Generalizing this method, one can construct in the unit interval nowhere dense sets of any measure less than 1, although the measure cannot be exactly one ( else its complement would be a nonempty open set with measure zero, which is impossible ).
40.
There are plenty of relevant countable dense sets of this space, e . g . polynomial functions with rational coefficients, or even simpler, piecewise affine functions with rational entries .-- talk ) 06 : 52, 27 August 2009 ( UTC)
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