
computably based locally compact
Lastmodified: 20100926 (日) 15:43:29
Definition
 A computably based locally compact space consists of a set of codes
for basic “points”, “open” and “compact” subspaces, together with an interpretation of
these codes in a locally compact sober space. We require of the space that every open
subspace be a union of basic ones. We also want to be able to compute
 codes (that we shall just call 0 and 1) for the empty set and the entire space, considered
as open and compact subspaces (if, that is, the entire space is in fact compact);
 codes for the union and intersection of two open subspaces, and for the union of two
compact ones, given their codes (we write + and ⋆ instead of ∪ and ∩ for these binary
operations, to emphasise that they act on codes, rather than on the subspaces that the
codes name);
 whether a particular representable point belongs to a particular basic open subspace,
given their codes; but we only need a positive answer to this question if there is one, as
failure of the property is indicated by nontermination;
 more generally, whether an open subspace includes a compact one, given their codes;
 codes for U and K such that L ⊂ U ⊂ K ⊂ V , given codes for L ⊂ V as above.
 In fact, we shall require the basic compact and open subspaces to come in pairs, with
U_n ⊂ K_n as in [JS96], where the superscript n names the pair, and we also need part
v. to yield such a pair as the interpolant.
Reference
 Taylor, Paul , Computably based locally compact spaces.[J] Log. Methods Comput. Sci. 2, No. 1, Paper 1, 70 p., electronic only (2006).
 [JS96] Jung, Achim and Sünderhauf, Philipp On the duality of compact vs. open.[A] Andima, Susan (ed.) et al., Papers on general topology and applications. Papers presented at the 11th summer conference at the University of Southern Maine, Gorham, ME, USA, August 1013, 1995. New York, NY: The New York Academy of Sciences. Ann. N. Y. Acad. Sci. 806, 214230 (1996)
