Effective Polish space

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. In particular, standard examples of Polish spaces such as the real line, the Cantor set and the Baire space are all effective Polish spaces.

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 ${\displaystyle \mathbb {N} ^{4}}$ computable (Moschovakis 2009:96-7):

${\displaystyle P(i,j,k,m)\equiv \left\{d(c_{i},c_{j})\leq {\frac {m}{k+1}}\right\}}$

${\displaystyle Q(i,j,k,m)\equiv \left\{d(c_{i},c_{j})<{\frac {m}{k+1}}\right\}}$

References

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