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**1 - 5**of**5**### SOME REMARKS ON COMPUTABLE (NON-ARCHIMEDEAN) ORDERED FIELDS

"... Let 2F and Jf be ordered fields such that & is a sub-field of Jf. JT is said to be Archimedean over ^ if, for any aeK, there exists a (leF such that a < /?. When ^ is the field of rational numbers SI and Jf is Archimedean over Si then one simply says that Jf is an Archimedean ordered field. $ ..."

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Let 2F and Jf be ordered fields such that & is a sub-field of Jf. JT is said to be Archimedean over ^ if, for any aeK, there exists a (leF such that a < /?. When ^ is the field of rational numbers SI and Jf is Archimedean over Si then one simply says that Jf is an Archimedean ordered field. $F is said to be dense in X if, for any a, ft e K such that a < /?, there is a y e F such that a < y < /?. Clearly, if OF is dense in Jf then X is Archimedean over 3F. For ^ =.2 the converse is also true; however, in general the converse is false. Indeed, one can give an example of an ordered field # &quot; such that # &quot; is not dense in its real-closure, say #. On the other hand, for any ordered field 2F, its real-closure # &quot; is Archimedean over &'. See [7]. In [4], we showed that if Jf is a computable (Archimedean) ordered extension of SI then Jf is isomorphic to a subfield of & c, the field of recursive real numbers. (In this paper we choose to use Si c for! % c, thus denoting real numbers whose rational cuts are recursive.) (1) Si c turns out to be the smallest extension field of Si which contains all computable (Archimedean) ordered extensions of Si. Equivalently, (2) SL C is the smallest extension field of Si which contains each computable ordered

### 12345efghi UNIVERSITY OF WALES SWANSEA REPORT SERIES

"... Computability on topological spaces via domain representations by V Stoltenberg-Hansen and J V Tucker Report # CSR 2-2007Computability on topological spaces via domain representations ..."

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Computability on topological spaces via domain representations by V Stoltenberg-Hansen and J V Tucker Report # CSR 2-2007Computability on topological spaces via domain representations

### Computability on topological spaces . . .

, 1997

"... Our aim in this thesis is to study a uniform method to introduce computability on large, usually uncountable, mathematical structures. The method we choose is domain representations using Scott-Ershov domains. Domain theory is a theory of approximations and incorporates a natural computability theor ..."

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Our aim in this thesis is to study a uniform method to introduce computability on large, usually uncountable, mathematical structures. The method we choose is domain representations using Scott-Ershov domains. Domain theory is a theory of approximations and incorporates a natural computability theory. This provides us with a uniform way to introduce computability on structures that have computable domain representations, by computations on the approximations of the structure. It is shown that large classes of topological spaces have satisfactory domain representations. In particular, all metric spaces are domain representable. It is also shown that the space of compact subsets of a complete metric space can be given a domain representation uniformly from a domain representation of the metric space. Several other classes of topological spaces are shown to have domain representations, although not all of them are suitable for introducing computability. Domain