Recursive Polish spaces

This paper is concerned with the proper way to effectivize the notion of a Polish space. A theorem is proved that shows the recursive Polish space structure is not found in the effectively open subsets of a space X, and we explore strong evidence that the effective structure is instead captured by the effectively open subsets of the product space N x X.

Automated summary

This paper discusses the proper approach to making the concept of a Polish space effective. A theorem is proven that demonstrates the absence of a recursive Polish space structure in the effectively open subsets of space X, and strong evidence is explored indicating that the effective structure is instead captured by the effectively open subsets of the product space N x X.