Math. Ann. 270, 83-86 (1985) lllathematische Annalen @ Springer-Verlag 1985 Finely Open Morphisms of /{-Cones Eugen Popa Department of Mathematics, University "A1. I. Cuza", R-6600 Iasi, Romania The main result of this note establishes the equivalence, under certain assumptions, of the following properties of a l{-map: - finely open, - finely or naturally locally non-constant, preserving the (semi-) polar sets under inverse image. One gets thus an effective generalisation of some previous results (on harmonic morphisms between harmonic spaces), obtained by: Constantinescu Cornea, Fuglede, Oja. Specifically, the following hypotheses are abandoned: the base space is locally compact, the natural sheaf property and the natural continuity of the considered morphism. All the notions and notations about .F/-cones are as in [2]. Moreover, the following definitions are used: If S and T are H-cones of functions on the sets X and Y, then the function q: Y--+ X is called a H-map if, for any bounded s € S, s o E belongs to ? (cf. [S]). Let S be a standard -E[-cone of functions on the nearly saturated space X. If GgX is a finely open set, then 9'(G) denotes (cf. Ul) the set of functions /: G--lR*, which are finite on a hnely dense subset, and for any finely open set D, with the fine closure contained in G, we have: flre 9(D). The author wishes to thank Prof. N. Boboc for helpful comments and advice. The Main Result Theorem l. Let S and T be standard H-cones of functions on the nearly saturated spaces X and Y. Let <p: Y"+ X be a H-map . Suppose that the following hypotheses are fulJilled: (i) X ls a Suslin space. (11) the points of X (or only q(Y)) are totally thin. (iii) E(f is a Borel set. (iv) .for any naturally open set U 9X and any bounded f e g'(U), we'haue: f " q e.q'(E-'(U)). The following properties for E are then equiualent: l) E is aJinely openmap. 2) VAgX totally thin +E- 11e; is totally thin. 3) Vx e X, ,p-t({*}) is nowhere finely dense in Y.