The Oscillation Mapping and the Square-bracket Operation

Part of the Progress in Mathematics book series (PM, volume 263)


In what follows, θ will be a fixed regular infinite cardinal.
$$ osc:\mathcal{P}\left( \theta \right)^2 \to Card $$
is defined by
$$ osc\left( {x,y} \right) = \left| {x\backslash \left( {\sup \left( {x \cap y} \right) + 1} \right)/ \sim } \right|, $$
where ∼ is the equivalence relation on x\ (sup(xy)+1) defined by letting αβ iff the closed interval determined by α and β contains no point from y. Hence, osc(x, y) is simply the number of convex pieces the set x \ (sup(xy)+1) is split by the set y (see Figure 8.1). Note that this is slightly different from the way we have defined the oscillation between two subsets x and y of ω1 in Section 2.3 above, where osc(x, y) was the number of convex pieces the set x is split by into the set y \ x. Since the variation is rather minor, we keep the same old notation as there is no danger of confusion. The oscillation mapping has proven to be a useful device in various schemes for coding information. Its usefulness in a given context depends very much on the corresponding ‘oscillation theory’, a set of definitions and lemmas that disclose when it is possible to achieve a given number as oscillation between two sets x and y in a given family X. The following definition reveals the notion of largeness relevant to the oscillation theory that we develop in this section.


Limit Point Vector Space Versus Bilinear Mapping Regular Cardinal Oscillation Mapping 
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© Birkhäuser Verlag AG 2007

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