Amoeba order

In mathematics, the amoeba order is the partial order of open subsets of 2^ω of measure less than 1/2, ordered by reverse inclusion. Amoeba forcing is forcing with the amoeba order; it adds a measure 1 set of random reals.

There are several variations, where 2^ω is replaced by the real numbers or a real vector space or the unit interval, and the number 1/2 is replaced by some positive number ε.

The name "amoeba order" come from the fact that a subset in the amoeba order can "engulf" a measure zero set by extending a "pseudopod" to form a larger subset in the order containing this measure zero set, which is analogous to the way an amoeba eats food.

The amoeba order satisfies the countable chain condition.

References

• Kunen, Kenneth (2011), Set theory, Studies in Logic, vol. 34, London: College Publications, ISBN 978-1-84890-050-9, MR 2905394, Zbl 1262.03001