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A topology on a space $X$ is defined as a subset of the power-set of X, that is closed under arbitrary unions, finite intersections and includes the empty set and the full space.

Is anybody aware of a modification of the notion of topology where closure is only under finite intersection and recursively enumerable collections of open sets?

Example: Consider the set of natural numbers, and let the set of open sets be the set of recursively enumerable sets of the natural numbers. Now for any finite collection of r.e. sets their intersection is also r.e. Furthermore, for an r.e. collection of sets, their union is r.e.

Has any work been done on a notion similar to this (if indeed that makes sense)?

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3 Answers 3

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In the setting of effective Polish spaces, you are describing the lightface $\Sigma^0_1$ pointclass. These are well-studied in effective descriptive set theory. The canonical reference on this subject is Moschovakis's Descriptive Set Theory, which is currently available online by the author.

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Thanks for the input! I read in computable topology and indeed it was not what i described. –  Mohamed Alaa El Behairy Jul 12 '11 at 4:17

The answer to "Has work been done on $X$?" is usually yes. If you Google "computable topology" you will get a fair number of hits.

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Hm. The first sentence is clearly false. Furthermore, computable topology (in the sense of Weihrauch) has nothing to do with the setup in the question. In computable topology, open sets are defined as usual but the functions between spaces are required to be computable with respect to a systems of notations for the spaces involved. –  François G. Dorais Jul 11 '11 at 19:05
    
I thought that indeed they used r.e. collections of open sets. –  André Nicolas Jul 11 '11 at 19:21
    
They consider codes for open sets but the codes do not have to be r.e. The codes are simply treated as oracles. So Weirauch style analysis is quite different than "Markov style" where everything is computable or r.e. –  Carl Mummert Jul 11 '11 at 22:25

Having finite joins and intersects gives us merely a lattice. Why would you define these as open, rather than as closed, for example?

The most interesting features of topologies are the ones that arise from the assymmetry of finite intersections/infinite joins.

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There is an asymmetry, the definition above requires closure over recursively enumerable unions and finite intersections , not finite unions and intersections. –  Mohamed Alaa El Behairy Jun 14 '11 at 6:03
    
I suspect the OP means "closure" in the sense of "stable under [an operation]". –  Zhen Lin Jun 14 '11 at 10:10

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