#### Glasnik Matematicki, Vol. 47, No. 1 (2012), 1-20.

### LOCAL COMPUTABILITY OF COMPUTABLE METRIC SPACES AND COMPUTABILITY OF CO-C.E. CONTINUA

### Zvonko Iljazović

Department of Mathematics, University of Zagreb, 10 000 Zagreb, Croatia

*e-mail:* `zilj@math.hr`

**Abstract.** We investigate conditions on a computable metric space
under which each co-computably enumerable set satisfying certain
topological properties must be computable. We examine the notion
of local computability and show that the result by which in a
computable metric space which has the effective covering property
and compact closed balls each co-c.e. circularly chainable
continuum which is not chainable must be computable
can be generalized to computable metric spaces which have the effective
covering property and which are locally compact. We
also give examples which show that neither of these two
assumptions can be omitted.

**2010 Mathematics Subject Classification.**
03D78.

**Key words and phrases.** Computable metric space, computable set, co-c.e. set,
local computability, the effective covering property.

**Full text (PDF)** (free access)
DOI: 10.3336/gm.47.1.01

**References:**

- V. Brattka,
*Plottable real number functions and the computable graph theorem,*
SIAM J. Comput. **38** (2008), 303-328.

MathSciNet
CrossRef

- V. Brattka and G. Presser,
*Computability on subsets of metric spaces*, Theoret. Comput.
Sci. **305** (2003), 43-76.

MathSciNet
CrossRef

- C. E. Burgess,
*Chainable continua and indecomposability,* Pacific J. Math. **9** (1959), 653-659.

MathSciNet
CrossRef

- C. O. Christenson and
W. L. Voxman, Aspects of topology, Marcel Dekker, Inc., New York, 1977.

MathSciNet

- Z. Iljazović,
*Chainable and circularly chainable co-r.e. sets
in computable metric spaces*, Journal of Universal Computer
Science **15(6)** (2009), 1206-1235.

MathSciNet

- J. S. Miller,
*Effectiveness for embedded spheres and balls,* Electron. Notes Theor. Comput. Sci. **66** (2002), 127-138.

CrossRef

- S. B. Nadler, Continuum theory. An introduction, Marcel Dekker, Inc., New York, 1992.

MathSciNet

- M. B. Pour-El and I. Richards, Computability in analysis and physics, Springer-Verlag, Berlin-Heielberg-New York, 1989.

MathSciNet

- E. Specker,
*Der Satz vom Maximum in der rekursiven Analysis,* Constructivity in Mathematics (A. Heyting, ed.), North Holland Publ. Comp., Amsterdam, 1959, 254-265.

MathSciNet

- K. Weihrauch, Computable analysis. An introduction, Springer, Berlin, 2000.

MathSciNet

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