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In many location problems, the solution is constrained to lie within a closed set. In this paper, optimal solutions to a special type of constrained location problem are characterized. In particular, the location problem with the solution constrained to be within a maximum distance of each demand point is considered, and an algorithm for its solution is developed and discussed.

From an arbitrary convergent sequence of sets of bounded variation we can select a subsequence such that there is convergence in almost every hyperplane.

In this paper a converse of Barwise's completeness theorem is proved by cut-elimination considerations applied to inductive definitions. We show that among the transitive sets T satisfying some weak closure conditions (closure under primitive-recursive set-functions is more than enough), only the unions of admissible sets satisfy Barwise's completeness theorem in the form stating that if φ ∊ T is a sentence which has a derivation (in the universe) then φ has a derivation in T. See §1 for the origin of the problem in Barwise's paper [Ba].Stated quite briefly the proof is as follows (a step-by-step account including relevant definitions is given in the body of the paper):Let T be a transitive prim.-rec. closed set, and let is nonempty, transitive and closed under pairs}. For each let κ(A) be the supremum of closure ordinals of first-order positive operators on subsets of A (first-order with respect to By Theorem 1 of [BGM], it is enough to prove that rank(T) in order to obtain that T is a union of admissible sets. (The rank of a set is defined by rank(x) = supy ∊ x (rank(y) + 1); since T is prim.-rec. closed, rank(T) = smallest ordinal not in T.)Let We show how to find in T (in fact, in Lω(A)) a derivable sentence τ that has no derivation D such that rank(D) ≤ α. Thus, if τ is to have a derivation in T, rank(T) > α. α is arbitrary (< κ(A)), so rank(T) ≥ κ(A). Q.E.D.

We examine encounter evasion game problems for a linear controlled system described by differential equations with a small parameter in a part of the derivatives [1], On the basis of the procedure of control with a leader [2, 3] we construct a strategy which ensures an encounter or an evasion generated by its motions relative to a specified closed target set within the limits of another closed set of phase coordinates. In particular, we examine the problem of evasion during an arbitrarily large time interval. The work relies on the formalization of differential games given in [2], As an example we consider the evasion problem for a system asymptotic with respect to the small parameter to a system described in [4].

We introduce the new topology on a topological space generated by the -sets. For an extensive subcategory of a category A of Hausdorff spaces and continuous maps, we consider the subcategory of A determined by those members of A which are -closed in their -reflection spaces. It is shown that is also an extensive subcategory of A for every infinite cardinal number if A is hereditary. It is also shown that a completely regular space is -closed in its Stone-Cech compactification iff it is -compact, that a zero-dimensional space is -closed in its maximal zero-dimensional compactification iff every maximal open closed filter with the -intersection property is fixed, that a Hausdorff uniform space is -closed in its completion iff every Cauchy filter with the -intersection property is convergent, and that a Hausdorff space is -closed in its Katětov extension iff every maximal open filter with the -intersection property is convergent.

By modifying the acceptability conditions in finite automata, a new and equivalent variant—the "structure automaton"— is obtained. The collection SR(Σ) of sets of tapes on Σ definable by deterministic structure-automata forms, however, a proper subset of the collection of regular sets. The structure and closure properties of SR(Σ) are analyzed in detail, using a natural topology on Σ*, in which the closed sets are the reverse ultimately definite sets. A set of tapes V is in SR(Σ) iff it is a finite union of regular "convex" sets. SR(Σ) is closed under Boolean operations, but not-closed under product, star, or transpose operations. In fact, SR(Σ) is exactly the Boolean closure of the regular closed sets. The "sigture" of a set is also defined and it is shown that a regular V is in SR(Σ) iff it has finite signature. Decision problems are also treated.

Using Scarf's algorithm for “computing” a fixed point of a continuous mapping, the following is proved: LetM 1 ⋯ M n be closed sets inR n which cover the standard simplexS, so thatM i coversS i , the face ofS opposite vertexi. We say a point inS iscompletely labeled if it belongs to everyM i andk-almost-completely labeled if it belongs to all butM k . Then there exists a closed setT ofk-almost-completely labeled points which connects vertexk with some completely labeled point. This result is used to prove Browder's theorem (a parametric fixed-point theorem) inR n . It is also used to generate “algorithms” for the nonlinear complementarity problem which are analogous to the Lemke—Howson algorithm and the Cottle—Dantzig algorithm, respectively, for the linear complementarity problem.

It is shown that the class F of compact T1 spaces is the union of classes F μ, one for each initial ordinal ωμ, where each subclass F μ is generated by a single element Wμ; the class F μ consists of the extension closed subspaces of products of Wμ, where a subspace is extension closed if every open cover of it extends to an open cover of the containing space. The space Wμ have the property that every descending chain of closed subsets is finite. Certain useful subspaces of the Wμ's are considered. The properties of various classes of spaces related to the universal spaces Wμ are considered, and certain cardinal invariants that can be attached to a space are examined.

In this paper we establish the existence of a best Lp approximation, 1 ⩽ p ⩽ ∞, to a given function f∈Lp( D , where D ⊂ Rm is a bounded domain, from the family Vn(S) of all nth order exponential sums in m independent variables for which the corresponding exponential parameters lie in the closed set S ⊆ C. In so doing we extend the previously known existence theorem which corresponds to the special case where m = 1 and D is a finite interval.

It is known that a lot of topological properties devolve from a basic space X X to the family M τ ( X ) {M_\tau }(X) of all τ \tau -smooth Borel measures endowed with the weak topology (or to certain subspaces of M τ ( X ) {M_\tau }(X) ). The aim of this paper is to show that among these topological properties there cannot be properties which are hereditary on closed subsets but not on countable products of X X , e.g. normality, paracompactness, the Lindelöf property, local compactness and σ \sigma -compactness. For this purpose it is proved that the countable product space X N {X^N} is homeomorphic to a closed subset of M τ ( X ) {M_\tau }(X) . A further consequence of this result is for example that, for the family M τ 1 ( X ) M_\tau ^1(X) of probability measures in M τ ( X ) {M_\tau }(X) , compactness, local compactness and σ \sigma -compactness are equivalent properties.

' In a semilattice two necessary and sufficient conditions for the interval topology to be compact are established. One is in terms of the fixed point property for increasing functions on the semilattice, and the other is in terms of completeness of the semilattice. Several years ago L. E. Ward, Jr. stated a characterization of the fixed point property for increasing functions on semilattices [5]. Subsequently the authors of the present paper detected a flaw in the argument. Since the theorem is of some importance [2] and has not been independently subsumed in the literature, it seems desirable to publish a corrected and improved version of the theorem. Let (X, ) are semilattices, then (X

This correspondence provides theorems for the existence of a nontrivial closed parition on the state set of a sequential machine and offers an efficient and systematic method for generating all such partitions. The method is based on the concept of the implication or closure dependence relation among some subsets of the state set.

The existence is proved of a solution of the equation xɛ F(t, x), where F(t, x) is a nonempty closed set depending continuously on t and x, with no assumptions concerning the convexity of this set.

In this paper we derive conditions necessary for a curve to minimize a functional of the Bolza form, subject to differential equation side conditions and the restriction that the trajectory of an admissible curve is constrained to lie inside a closed set S with a smooth boundary. These results are obtained as limits of conditions derived by E. J. McShane for curves whose trajectories are contained in a neighborhood of S. This limit procedure verifies the continuity of optimal solutions as a “soft” boundary of S “hardens” until it allows no penetration at all.

The propertyPm (directly analogous to Valentine’s propertyP3) is used to prove several curious results concerning subsets of a topological linear space, among them the following: (a) If a closed setS has propertyPm and containsk points of local nonconvexity no distinct pair of which can see each other viaS, thenS is the union ofm − k − 1 or fewer starshaped sets. (b) Any closed connected set with propertyPm is polygonally connected. (c) A closed connected setS with propertyPm is anLm−1 set (each pair of points may be joined by a polygonal arc ofm − 1 of fewer sides inS). (d) A finite-dimensional set with propertyPm is anL2m − 3 set. A new proof of Tietze’s theorem on locally convex sets is given, and various examples refute certain plausible conjectures.

If X is a compact Hausdorff space we denote by S(X), and by C(X), the hyperspaces of X consisting of all non-empty closed sets, and all non-empty connected closed sets. The topology in each case is the finite topology of Michael ((6), Definition 1·7), in which a sub-base for the open sets is taken consisting of all sets of either of the forms {F|F ⊂ G} and {F|F ∩ G ≠ φ} (where G is any open set of X). Michael has shown that S(X) is also compact Hausdorff ((6), Theorem 4·9·6), and S(X) contains in an obvious way sets which are homeomorphic with C(X) and X itself. We recall that if Xis also a metric space, the topology induced on S(X) (and on C(X)) by Hausdorff's metric is the same as the finite topology ((6), Proposition 3·6).

We consider an antagonistic pursuit game with a life-line with simple motions taking place in a given convex closed set in the plane. The evader E is considered to have been captured if the Euclidean distance $\rho (P,E)$, where P is the pursuer, does not exceed l, where l is some preassigned positive number. Optimal strategies which make up an equilibrium point are found for both players.

A space X is said to have property (B if for any well-ordered monotone decreasing family {Ha aEzA } of closed sets with no common part, there is a monotone decreasing family of domains {Dal aEA I such that (i) HaCDa for each a in A and (ii) {cl(Da)f IC A } has no common part. It is shown that property (B characterizes the separable T3-spaces that are Lindelof and the countably compact spaces that are compact. Also, it is shown that the T3-space X is Lindel6f if and only if X has property (B and every uncountable subset of X has a limit point. Throughout this paper, topological spaces are assumed to be Ti-spaces.

A Markov random set is a time-homogeneous random closed set on the half-line $t \geqq 0$, satisfying the Markov property of independence between the future and the past when the present is known. Such sets are introduced as a special class of Markov processes. They may be described by a non-increasing right-continuous positive function $g(x)$, $x > 0$, integrable near 0 and a non-negative number $\alpha $, determined up to an arbitrary positive constant factor. If $y(t)$ is a continuous strong Markov process, the t-set $\{ {y(t) = {\text{const}}} \}$ is a Markov random set. The most interesting Markov sets are obtained by simple transformations from the Brownien motion process.

Various conditions under which an upper semi-continuous (u.s.-c.) decomposition of E3 yields E3 as its decomposition space have been given by Armentrout (1; 2; 5), Bing (7; 8), Lambert (13), McAuley (14), Smythe (17), and Wardwell (18). If the projection of the non-degenerate elements is 0-dimensional in the decomposition space, then “shrinking” or “Condition B” (6) has proven particularly useful.In this paper we shall investigate monotone u.s.-c. decompositions of a locally compact connected metric space M, where the projection of the nondegenerate elements is 0-dimensional. We show in Theorem 1 that each open covering of the non-degenerate elements of a 0-dimensional decomposition has a locally finite refinement.In § 5, we use Theorem 1 to investigate the following question which is similar to one raised by Bing (11, p. 19): Let G, G′, and G″ be decompositions of M such that the non-degenerate elements of G are those of G′ together with those of G′.

We consider a Markov process $x_0, x_1, \cdots$ with a countable set $S$ of states and stationary transition probabilities $p(t \mid s) = P\{x_{n+1} = t \mid x_n = s\}$. Call a set $C$ of states almost closed if (a) $P\{x_n \varepsilon C$ for an infinite number of $n\} > 0$ and (b) $x_n \varepsilon C$ infinitely often implies $x_n \varepsilon C$ for all sufficiently large $n$, with probability one. It is shown that there is a set $(C_1, C_2, \cdots)$ essentially unique, of disjoint almost closed sets such that (a) all except at most one of the $C_i$ are atomic, that is, $C_i$ does not contain two disjoint almost closed subsets, (b) the non-atomic $C_i$, if present, contains no atomic subsets, (c) the process is certain to enter and remain in some set $C_i$. A relation between the sets $C_i$ and the bounded solutions of the system of equations \begin{equation*}\tag{1} \alpha(s) = \sum_t \alpha(t)p(t \mid s)\end{equation*} is obtained; in particular there is only one atomic $C_i$ and no non-atomic $C_i$ if and only if the only bounded solution of (1) is $\alpha(t)$ = constant. This condition is shown to hold if the process is the sum of independent identical (numerical or vector) variables; whence, for such a process, the probability of entering a set $J$ infinitely often is zero or one. The results are new only if the process has transient components. The main tool is the martingale convergence theorem.

Giving Ix the topology of uniform convergence, we have that the closure of a set with property V has property V, as does the intersection of such sets. Thus every subset of Ix is contained in a smallest set with property V, and in a smallest closed set with property V. If X is a topological space then the set D(X) of all continuous functions from X into I is closed and has property V. The idea of considering such collections of functions comes from a statement of von Neumann in [1]. Essentially, he claims without proof what we give here as a corollary to Theorem 2. I am indebted to Dr. R. S. Pierce for bringing the problem to my attention. DEFINITION. If n is a positive integer, let P,n be the smallest subset of D(In) that has property V and contains the n projections.

Y is a Wallman-type compactification (O. Frink, Amer. J. Math. 86 (1964), 602-607) of X in case there is a normal base Z for the closed sets of X such that the ultrafilter space from Z, denoted ω ( Z ) \omega (Z) , is topologically Y. It is not known if every compactification is Wallman-type. For Z α {Z_\alpha } a normal base for the closed sets of X α {X_\alpha } for each a belonging to an index set Δ \Delta it is shown that the Tychonoff product space ∏ α ∈ Δ ω ( Z α ) {\prod _{\alpha \in \Delta }}\omega ({Z_\alpha }) is a Wallman compactification of ∏ α ∈ Δ X α {\prod _{\alpha \in \Delta }}{X_\alpha } . Also for X ⊂ T ⊂ ω ( Z ) X \subset T \subset \omega (Z) with Z a normal base for the closed sets of X, a proof that ω ( Z ) \omega (Z) is a Wallman-type compactification of T is indicated.

This paper contains proofs of the closure of certain sets of rational functions in various spaces. Thus, for example, conditions are derived for the closure of the sequence (x2 + zv2)−1 in the space L2(0, ∞), and for the set\(\frac{{ct - z_v }}{{1 - \bar ctz_v }}\) in C(−1, +1). Analogous results are proved for other related sets of rational functions. Some of these results are new; others are new proofs of known theorems. The main point is that a uniform method is used throughout this paper. For a description of the method see article 1.

The similarity of physical processes described by elliptic equations in stationary problems of diffusion and by the single-energy transfer equation in the theory of radiation suggests that some mathematical properties of these equations must be similar. The following circumstances mast be borne in mind when the principles of maximum and minimum are extended to the transfer equation. The solution of ordinary boundary problems for the equation in the sense of intensity of radiation cannot be considered continuous in the domain of definition since, for example, any discontinuity in the boundary conditions is propagated along rays of emerging from the point of discontinuity. On the other hand, the domain of definition of the solution is not a closed set. At points of the boundary, in particular, it is not always possible to determine the intensity in tangential directions. For this reason the methods of proof developed in the theory of elliptic equations cannot be used directly when the transfer equation is studied. It is however possible to prove a proposition sufficient for formulating the usual corollaries of the principles of maximum and minimum: uniqueness of the solution of the boundary problem, continuous dependence of the solution on boundary conditions etc. (§ 1). The possibility of the solution attaining maximum or minimum values at various points in the domain of definition is investigated in § 2.

An optimal control problem is considered in which it is desired to transfer a linear control system from one given state to another state. The target state may either be a point or a convex closed set. Optimization is understood in the sense of minimizing the control effort, where effort is defined either as maximum amplitude or as an integral of a certain function of the control. The optimization problem is reduced to the problem of finding the unique minimum of a function of n variables (where n is the order of the system) . It is shown that the method of steepest descent is particularly applicable to finding this minimum, and consequently to determining the minimum effort and optimal control.

Valentine (3) introduced the three-point convexity property P3 : a set S in En satisfies P3 if for each triple of points x, y, z in S at least one of the closed segments xy, yz, xz is in S. He proved, (3 or 1) that in the plane a closed connected set satisfying P3 is the union of some three convex subsets. The problem of characterizing those sets that are the union of two convex subsets was suggested. Stamey and Marr (2) have provided an answer for compact subsets of the plane. We present here a generalization of property P3 which characterizes closed sets in an arbitrary topological linear space which are the union of two convex subsets.

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We investigate the approximating properties of sets in spaces normed over semifields. We establish criteria for sets to possess the minimization property in several classes of spaces. A set possesses this property if, for every element of the space, there is a sequence of elements of the set which give arbitrarily close approximations to the best approximation. 5 titles.

Let Y be a Hausdorff compactification of a locally compact space X and let K = Y − X K = Y - X be the remainder. Suppose that K is a regular Wallman space (i.e. K has a base for closed sets which is a ring and which consists of sets each of which is the closure of its interior) and suppose that K is a neighborhood retract of Y. Under these assumptions, it is proved below that Y is a Wallman compactification of X.

In a recent paper [6] Wermer considered the algebra C of all continuous complex valued functions on y, a simple closed anal-ytic curve bounding a region r, with ruy compact, on a Riemann surface F. He considered the subalgebra A of all functions in C which could be extended into r to be analytic on r and continuous on ruJy. Wermer showed that A was a maximal closed subalgebra of C which separated the points of y, and that the space of maximal ideals of A was homeomorphic to ruJy. In [2 ] Civin and Yood considered a class of subalgebras of complex commutative regular Banach algebras which become maximal closed subalgebras in the event the original algebra was the collection of continuous functions on a compact Hausdorff space. The object of this note is to demonstrate that such subalgebras possess a maximum modulus property possessed by A. To state the result obtained we recall certain definitions. The terms not herein defined may be found in [5]. Let B be a complex commutative regular Banach algebra with identity e and space of maximal ideals 9M(B). Let r: x-*x(M) be the Gelfand representation of B as a subalgebra of C((B)), the continuous function on 9M(B). We also denote irx by 2 and xrQ by Q for any subset Q of B. A subalgebra N of B is called determining [2] if irN is dense in irB, otherwise N is called nondetermining. A subalgebra of B is called a maximal nondetermining subalgebra if every larger subalgebra of B is deternining. A subset S of B is called a separating family on 2Z(B) if for each M1, M2 in 9YI(B), M1lM2, there exists an xES such that x(Mi) $x(M2). If P is an algebra of continuous complex valued functions vanishing at infinity on the locally compact space X, the smallest closed set (if it exists) on which each |ft with fGP assumes its maximum is called the gilov boundary of X with respect to P.

w 1. Introduction What point sets in the complex plane contain infinitely many sets of conjugate algebraic integers ? A basic contribution to this question was made by FEKETE [1], who showed that a bounded closed set E with transfinite diameter less than 1 can contain only a finite number of such sets of conjugates. The strict converse is not true, but a substitute is furnished by FEKETE and SZEG(J [2], Theorem K, which states that if E is symmetric to the real axis and the transfinite diameter of E is at least 1, then any open set D including E will contain infinitely many sets of conjugate algebraic integers. We may say briefly that there are infinitely many sets of conjugate algebraic integers near E. (An analogous theorem about real point sets was proved in [5].) These results may be combined in the statement that, for a bounded closed set E which is symmetric to the real axis, there are infinitely many sets of conjugate algebraic integers near E if and only if the transfinite diameter of E is at least 1. The condition that E is bounded may be dropped, if we define the terms properly. When we speak of an unbounded set E being closed, this will be understood in the spherical sense; that is, E must contain the point at infinity. Any open set D containing E will then contain all points outside of some circle, and hence will certainly contain infinitely many sets of conjugate algebraic integers. By definition, the transfinite diameter of an unbounded closed set is infinite.

In a recent paper (see [2]), Orrin Frink introduced a method to provide Hausdorff compactifications for Tychonoff or completely regular T1 spaces X. His method utilized the notion of a normal base. A normal base ℒ for the closed sets of a space X is a base which is a disjunctive ring of sets, disjoint members of which may be separated by disjoint complements of members of ℒ.

We investigate the statistical properties of the intensity of radiation emitted by $N$ two-level systems, for which the cooperation number is constant. The $l$-fold moments of the intensity are shown to obey a closed set of equations. For long times, the normalized moments are found to go up to infinity. But, taking into account the experimental limitations, large fluctuations will be seen only for almost completely excited systems.

In [3] Orrin Frink introduced the notion of a Wallman compactification of a Tychonoff space. A. K. and E. F. Steiner [6] have recently shown that any compact metric space (or product of compact metric spaces) is a Wallman compactification of each of its dense subspaces. Their result is an immediate consequence of earlier work and a theorem they proved about the existence of a certain kind of base of closed sets in a compact metric space. The purpose of this note is to give a different proof of this latter result, obtaining as a consequence a slightly stronger version. DEFINITION. Let X be a Tychonoff space. A family Z of closed subsets of X is said to be a regular normal base for X if z is a base for the closed subsets of X, if each member of Z is a regular set (the closure of its interior) and if (i) Z is closed under finite unions and intersections; (i.e., Z is a ring) (ii) if xEX, F a closed subset of X, xTF, then there exists ZEZ such that xEEZCX\F; and (iii) if A, BE Z, A nB=0 then there exists C, D E Z such that

In this paper a mapping A of a closed subset D of a Hausdorff space E into E is considered, and the set N(A) of solutions of the equation x = Ax on a closed set D'⊂D is studied. Bibliography: 20 items.

In this paper we shall investigate the so-called weak convergence of measures. Although the origin of the concept of the weak convergence of measures is a probabilistic one, the concept itself is purely measure-theoretical, and should be, therefore, treated by measure-theoretical methods. In Probability Theory the notion of the weak convergence of measures first appeared in Central Limit Problem. Its full importance, however, has been recognized only recently. It is now known as Donsker's Invariance Principle. In this paper we shall follow Prohorov's approach, as presented in [1]. The list of all necessary definitions and results is given in the Introduction. We shall give some conditions for the weak convergence of measures in separable and complete metric spaces, which are expressed in terms of convergence of measures generated in finite dimensional Euclidean spaces. The last convergence can be treated by standard mathematical tools, like the Theory of Fourier Transformations. It should be noted that our theorems concerning the convergence of measures in separable complete metric spaces remain valid if we omit the assumption of completeness. The proofs will remain essentially unchanged; only instead of dealing with compact sets, we should deal with totally bounded closed sets. The theorems given in Section 4 are of interest for the Theory of Stochastic Processes, since they give the conditions for the weak convergence of measures in the functional spaces $D\lbrack 0, 1\rbrack$ and $C\lbrack 0, 1\rbrack$, and to a large class of stochastic processes there correspond measures generated in space $C\lbrack 0, 1\rbrack$, and these measures are usually given in terms of $\mu^{t_1, \cdots, t_m}$, i.e. in terms of finite dimensional distribution functions of the process.

This paper is concerned with certain aspects of acyclicity in a compact connected topological semigroup, and applications to the admissibility of certain multiplications on continua. The principal result asserts that if S is a semigroup on a continuum, finitely floored in dimension 2, then S=ESE implies S=K. Introduction. We are concerned here with some aspects of acyclicity in a compact connected semigroup, and applications to the admissibility of a semigroup structure on "finitely floored" spaces (Definition 2). The main result is that if S is a semigroup on a continuum, finitely floored in dimension 2, then S=ESE implies S= K. This is a generalization of the result of Cohen and Koch [1], where the same conclusion is obtained assuming that S is a floor for each nonzero h E H2(S), cd S_ 2, and S is locally Euclidean except possibly at one point. A preliminary result which may be of interest in itself is that if S is a semigroup with a zero satisfying S =ESE on a continuum, then for each nonzero h E H2(S) there exists a pair of idempotents e andf such that hiSe u Sf# O. The notation is that of [4]. In particular, S denotes a topological semigroup, K the minimal ideal, and E the set of idempotents. For a closed set A of S, S/A denotes the space obtained by shrinking A to a point. The cohomology used is that of Alexander-Wallace-Spanier with coefficient group arbitrary. Throughout the paper we shall use reduced groups in dimension 0. We denote by A\B the complement of B in A, the closure of A by A* and the empty set by 1i. If Hn(A). The following theorem is due to Wallace [5]: THEOREM 1. Let X and Y be compact Hausdorff spaces and < a closed relation from X to Y such that L(A) r) L(B) is connectedfor each pair of closed subsets A and Received by the editors June 29, 1970. AMS 1970 subject classifications. Primary 22A15; Secondary 22A15.

In my paper Concerning non-dense plane continuat I showed that if in the plane S the set M is the sum of a countable number of closed sets containing no domain then there exists a topological transformation 11 of the plane S into itself such that if L is any straight line whatsoever the point set L -H(M) is totally disconnected. The principal object of the present paper is to prove this result with "plane S" replaced by "euclidean space of n dimensions." In the proof here given use is made of a general theorem concerning transformations in a locally compact, complete metric space.

Let G be a locally compact abelian group, and A ( G ) A(G) the Fourier algebra on G. A Helson set in G is called an SH-set if it is also an S-set for the algebra A ( G ) A(G) . In this article we prove that a compact subset K of G is an SH-set if and only if there exists a positive constant b such that: For any disjoint closed subsets K 0 {K_0} and K 1 {K_1} of K, we can find a function u in A ( G ) A(G) such that ‖ u ‖ > b , u = 1 \left \| u \right \| > b,u = 1 on some neighborhood of K 0 {K_0} , and u = 0 u = 0 on some neighborhood of K 1 {K_1} .

We shall first give some definitions concerning parametric surfaces. Denote by H a closed circle (disk) and by M a variable point on it. Let P = Ф(M) be a continuous function on H whose value P is a point in three-dimensional space. The symbols Ф(E), Ф−1(P), where E is a set of points on H and P a point in the three-dimensional space, will have their usual meaning. Ф−1(P) is a closed set. Any saturated continuum in Ф−1(P) or any point of Ф−1(P) that does not belong to such continua is called a Ф-element of H. Thus to any continuous function Ф(M) corresponds a representation of H in the form of the sum σQ of Ф-elements. The set of the pairs (P, Q), where Q runs through all Ф-elements of H and, for any Q, P = Ф(Q), is called a parametric surface, and any pair (P, Q) is called a point of the parametric surface. We shall often speak of a point Ф(M) of the parametric surface, by which we shall mean either the point (P, Q), where P = Ф(M) and Q is the Ф-element containing M, or the point P = Ф(M) of the three-dimensional space. The exact meaning will always be clear from the context. If there are exactly k points of the parametric surface whose first member is P0 we say that P0 is a point of multiplicity k. If k = 1, P0 is a simple point.