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 N4{displaystyle mathbb {N} ^{4}}{mathbb {N}}^{4} computable (Moschovakis 2009:96-7):



P(i,j,k,m)≡{d(ci,cj)≤mk+1}{displaystyle P(i,j,k,m)equiv left{d(c_{i},c_{j})leq {frac {m}{k+1}}right}}{displaystyle P(i,j,k,m)equiv left{d(c_{i},c_{j})leq {frac {m}{k+1}}right}}

Q(i,j,k,m)≡{d(ci,cj)<mk+1}{displaystyle Q(i,j,k,m)equiv left{d(c_{i},c_{j})<{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. .mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
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