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Complex analysis deals with holomorphic mappings between complex spaces. We ought to allow complex spaces to be infinite-dimensional and to have singularities. There are well-developed theories in the finite-dimensional case (see e.g. [GR], [Lo]) and for domains in certain kinds of topological vector spaces (see e.g. [Di], [FV], [He]). The most obvious approach to infinite-dimensional complex spaces is too general to yield sensible results [Do], but there has been progress [Au] in the study of (possibly infinite-dimensional, possibly singular) semi-Fredholm-analytic spaces. In any case, I shall assume that a complex space carries the structure of a Hausdorff space, which I shall call its underlying topology ; in the finite-dimensional case I shall assume that the underlying topology is paracompact. This note discusses some foundational questions and summarizes what we know about the relationships between the underlying topology and the topologies induced by the classical pseudodistances of Caratheodory and Kobayashi. Let Hol(X,Y ) denote the set of all holomorphic mappings from the complex space X into the complex space Y ; I shall assume that these mappings are continuous with respect to the underlying topologies of X and Y . Let D denote the open unit disc in C. A mapping f ∈ Hol(D, X) is called an analytic disc in X. I shall also assume that a complex space has the property that any two points can be connected by (the images of) a finite chain of analytic discs in the space; in the finite-dimensional case this is equivalent to connectedness in the topological sense [K70, pp. 97–98], [La, pp. 15–16].

The indicatrix of the Kobayashi infinitesimal metric at the center of a pseudoconvex complete circular domain coincides with this domain. It follows that a nonconvex complete circular domain cannot be biholomorphic to any convex domain. An example shows that a bounded pseudoconvex complete circular domain in C2 need not be taut. Let V be a complex Banach space with norm 11 11. A complete circular domain is a nonempty open set M C V such that XM C M for all X E C with i X I 0: v E XM} establish a one-to-one correspondence between the complete circular domains M and the semigauges p on V. Moreover: (a) M is bounded if and only if p is a gauge; (b) M is convex if and only if p is a seminorm; (c) M is pseudoconvex if and only if p is plurisubharmonic. In case V has finite dimension: (d) M is taut [10, p. 199] if and only if p is a continuous plurisubharmonic gauge. PROOF. Standard normed space techniques yield the one-to-one correspondence and properties (a) and (b). (c) If p is plurisubharmonic, it follows immediately [8, p. 42] that M is pseudoconvex. On the other hand, by definition [8, p. 41], M is pseudoconvex if and only if the function -log 8 is plurisubharmonic, where 8: M X (V {O}) -> (0, oo] is defined by 8(z,v) = inf l : X E C,z + Xv X M}. Received by the editors October 6, 1982. 1980 Mathematics Subject Classification. Primary 32H15; Secondary 32A07, 32F15.

In analogy with the Hartogs theorem that separate analyticity of a function implies analyticity, it is shown that a separately normal family of holomorphic maps from a polydisk into a Riemann surface is a normal family. This contrasts with examples of discontinuous separately analytic maps from a bidisk into the Riemann sphere. The proof uses a theorem on pseudoconvexity of normality domains, which is proved via the following convergence criterion: a sequence { f j } \{ {f_j}\} of holomorphic maps from a complex manifold into a Riemann surface converges to a nonconstant holomorphic map if and only if the sequence { f j − 1 } \{ f_j^{ - 1}\} of set-valued maps, defined on the Riemann surface, converges to a suitable set-valued map. Extending Osgood’s theorem, it is also shown that a separately analytic map (resp. a separately normal family of holomorphic maps) from a polydisk into a hyperbolic complex space is analytic (resp. normal).

Examples of the following are given: (i) complex spaces whose Caratheodory distances are not inner; (ii) a taut complex space that is not complete hyperbolic; (iii) a complex space that is complete hyperbolic mod A but not taut mod A; (iv) a bounded pseudoconvex domain that is not taut. This note provides negative answers to a few of the problems posed in S. Kobayashi's recent Bulletin survey [8]. All definitions, notation, and terminology follow that paper. In particular, p denotes the Poincare distance on the unit disk D, and cx (resp. dx) denotes the Caratheodory (resp. Kobayashi) pseudodistance on the (reduced) connected complex space X. 1. Inner pseudodistances. Let d be a pseudodistance on the topological space T. (We ignore the topology defined by d; it may have nothing to do with the given topology on T.) The length of a curve -y: [a, b] -* T is L(y) = sup , d (y(ti -1), y(ti)), where the supremum is taken over all partitions a = to < t1 < ... < tk = b of [a, b]; -y is called rectifiable if its length is finite. If T is finitely arcwise connected in the sense that each pair of points of T can be joined by a rectifiable curve, then a new pseudodistance d', called the inner pseudodistance induced by d, is defined by d' (p, q) = inf L(y), where the infimum is taken over all rectifiable curves -y joining p to q. (We emphasize that these curves are continuous as maps into T; we assume nothing about continuity with respect to d.) We say that d is inner if d = d'. A connected complex space X is finitely arcwise connected with respect to cx and dx. Moreover, it is easy to see that dx is inner [7, pp. 483-484]. In spite of his earlier assertion that cx is not always inner [7, p. 484], Kobayashi now asks whether this is true [8, Problem A. 5, p. 400]. In each of the following three examples, the distance cx is not inner. In the second example, X is a bounded domain of holomorphy; in the third, X is finitely compact with Received by the editors September 30, 1976. AMS (MOS) subject classifications (1970). Primary 32H15, 32H20; Secondary 32A17, 32F15.

A geometrically convex domain in C n {{\mathbf {C}}^n} is Kobayashi hyperbolic if and only if it contains no complex affine lines. This contrasts with an example of a nonhyperbolic pseudoconvex domain in C 2 {{\mathbf {C}}^2} containing no (nonconstant) entire holomorphic curves.

Taut and tight manifolds, introduced recently by H. Wu, are characterized as follows. Let D D denote the open unit disk in C C . The complex manifold N N is taut iff the set A ( D , N ) A(D,N) of holomorphic maps from D D into N N is a normal family. If d d is a metric inducing the topology on N , ( N , d ) N,(N,d) is tight iff A ( D , N ) A(D,N) is equicontinuous. It is also shown that every taut manifold is tight in a suitable metric.

The Kobayashi pseudodistance on a connected complex space is continuous with respect to the standard topology. IC this pseudodistance is an actual distance, it induces the standard topology. Let X be a connected (reduced) complex space. The Kobayashi pseudodistance dx(p, q) between points p and q of X is defined as follows ([6, p. 462], [7, pp. 97-98]). Let p denote the distance defined by the PoincareBergman metric on the open unit disk D in the complex plane. Choose points P PO,PloP' . Pk-1 pk=q of X, points a1,, * * , ak, bl, * * *, bk of D, and holomorphic maps fl, *, *fk from D into X such that fj(aj)= Pj-l and fj(bj)=pj for j= 1, , k. Then dx(p, q) is defined to be the infimum of the numbers p(a,, bl)+... + p(ak, bk) taken over all such finite chains joining p and q. Clearly dx is a pseudodistance on X. In case dx is an actual distance we will show that it induces the standard topology, i.e., the topology underlying the given complex structure on X. Several authors have tacitly used this fact; cf. [1, Proposition 3.8, pp. 68-69], [4, Proposition 1, pp. 50-51], [5, Theorems 2 and 3, pp. 590-591], [6, Theorem 3.4, pp. 465-466], [8, Theorem 1, pp. 11-12]. The only published proof, however, seems to be the one given by H. L. Royden [9, Theorem 2, pp. 133-134] for complex manifolds. Our proof uses only three elementary facts about the Kobayashi pseudodistance: if f: Y--Z is a holomorphic map, it is distance decreasing in the sense that dz(f(p),f (q))