Plateau principle

In mathematical logic, an effective Polish space is a complete separable metric space that has a computable presentation. Such spaces are studied in effective descriptive set theory and in constructive analysis.

Definition

An effective Polish space is a complete separable metric space X with metric d such that there is a countable dense set C = (c₀, c₁,...) that makes the following two relations on ${\mathbb{\mathbb{N}}}^4$ computable (Moschovakis 2009:142):

$P(i,j,k,m)\equiv d(c_i,c_j)\le \frac{m}{k+1}$

$Q(i,j,k,m)\equiv d(c_i,c_j)<\frac{m}{k+1}$

References

• Yiannis N. Moschovakis, 2009, Descriptive Set Theory, 2nd edition, American Mathematical Society. ISBN 0-8218-4813-5