We consider some semilinear (= semiaffine) and semialgebraic loci of o-minimal sets in euclidean spaces. Semilinear loci have good properties. Some of these properties hold for semialgebraic loci when we restrict to a smaller class of analytically o-minimal sets. 1. O-minimal sets and geometric loci A structure τ on the set of real numbers R is a family τn (n ∈ N) such that (cf. [4] or [5]): (S1) τn is a boolean algebra of subsets of R, (S2) if A ∈ τn, then R×A, A×R belong to τn+1, (S3) if A ∈ τn+1, then π(A) ∈ τn, where π : R → R is the natural projection obtained by dropping the last component, (S4) the diagonals {x ∈ R : xi = xj} for 1 ≤ i < j ≤ n belong to τn. If also (S5) singletons {r} for r ∈ R belong to τ1, (S6) the linear order {(x, y) ∈ R : x ≤ y} belongs to τ2, (S7) every set from τ1 is a finite union of intervals (of any type), then this structure is o-minimal. We will say that A ⊂ R (not necessary a proper subset of R) belongs to τ if A ∈ τn. The following two examples of o-minimal structures are widely known (see, for example, [5]): • The system of semilinear sets. • The system of semialgebraic sets. For a finite collection F of subsets of R,R,R, . . . , we define Tarski(F) to be the smallest structure on R containing F and the semialgebraic sets. We call F (or a single set A) o-minimal if Tarski(F) (or Tarski(A)) is o-minimal. In Section 3 we prove: Received November 15, 2000; received in final form May 21, 2001. 2000 Mathematics Subject Classification. Primary 03C64, 14P99. Secondary 32B20. c ©2001 University of Illinois