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| | In [[mathematical logic]], an '''effective Polish space''' is a [[complete metric space|complete]] [[separable space|separable]] [[metric space]] that has a [[computable presentation]]. Such spaces are studied in [[effective descriptive set theory]] and in [[constructive analysis]]. |
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| | == Definition == |
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| | An effective Polish space is a complete separable metric space ''X'' with metric ''d'' such that there is a countable dense set ''C'' = (''c''<sub>0</sub>, ''c''<sub>1</sub>,...) that makes the following two relations on <math>\mathbb{N}^4</math> computable (Moschovakis 2009:142): |
| | :<math>P(i,j,k,m) \equiv d(c_i,c_j) \leq \frac{m}{k+1}</math> |
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| | :<math>Q(i,j,k,m) \equiv d(c_i,c_j) < \frac{m}{k+1}</math> |
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| | == References == |
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| | *[[Yiannis N. Moschovakis]], 2009, ''Descriptive Set Theory'', 2nd edition, American Mathematical Society. ISBN 0-8218-4813-5 |
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| | [[Category:Computable analysis]] |
| | [[Category:Effective descriptive set theory]] |
| | [[Category:Computability theory]] |
| | {{mathlogic-stub}} |
Revision as of 22:03, 24 December 2013
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 computable (Moschovakis 2009:142):
References
- Yiannis N. Moschovakis, 2009, Descriptive Set Theory, 2nd edition, American Mathematical Society. ISBN 0-8218-4813-5
Template:Mathlogic-stub