On approximation of topological algebraic systems by finite ones

We introduce a definition of approximation of a topological algebraic system A by systems of a class K and investigate it for the case when the class K consists of finite systems. For the case when the topology on the system A is discrete this definition is equivalent to the well-known definition of local embedding of A in K, which means that A is a subsystem of an ultraproduct of some systems in the class K. We obtain a similar characterization of approximation of a locally compact system A by systems in K. We consider the bounded formulas in the signature of A and their approximations similar to those, introduced for Banach spaces in the paper of C.W.Henson ”Nonstandard hulls of Banach spaces” Israel Journal of Mathematics (1976), v.25, pp. 108 144. We prove that a positive bounded formula φ holds in A if all precise enough approximations φ hold in all precise enough approximations of A. We prove that a locally compact field cannot be approximated by finite associative rings (not necessary commutative). Finite approximations of the field R can be interpreted as computer systems for reals. So it is impossible to construct a computer arithmetic for reals that is an associative ring.