Showing papers on "Disjoint sets published in 1971"

Entropy and Molecular Correlation Functions in Open Systems. I. Derivation

Abstract: A method is presented for obtaining an expression for the entropy in terms of molecular correlation functions defined in the grand canonical ensemble. The procedure is for a system of a single molecular species whose dynamics are determined by classical equations of motion. The entropy is obtained as a sum of two different classes of functions each involving the correlations between n‐tuples of molecules. One class contains logarithmic terms similar to those obtained for the closed system; the other class involves isothermal activity derivatives of potentials of mean force. The latter terms, which are moments of the correlations between disjoint sets of molecules, can make appreciable contributions to the entropy. The method leads to results similar to those obtained from a different procedure by Nettleton and Green. The expression for the entropy is obtained and properties of the results are discussed for a simple fluid system.

Error Bounds for Approximate Invariant Subspaces of Closed Linear Operators

Abstract: Let A be a closed linear operator on a separable Hilbert space $\mathcal{H}$ whose domain is dense in $\mathcal{H}$ Let $\mathcal{X}$ be a subspace of $\mathcal{H}$ contained in the domain of A and let $\mathcal{Y}$ be its orthogonal complement. Let B and C be the compressions of A to $\mathcal{Z}$ and $\mathcal{Y}$ respectively, let $G = Y^ * AX$, where X and Y are the injections of $\mathcal{X}$ and $\mathcal{Y}$ into $\mathcal{H}$. It is shown that if B and C have disjoint spectra and $\| G \|$ is sufficiently small, then there is an invariant subspace $\mathcal{X}'$ of A near $\mathcal{X}$. Bounds for the distance between $\mathcal{X}'$ and $\mathcal{X}$ are given, and the spectrum of A is related to the spectra of B and C. In the development a measure of the separation of the spectra of B and C which is insensitive to small perturbations in B and C is introduced and analyzed.

A class of ergodic transformations having simple spectrum

Abstract: A class of ergodic, measure-preserving, invertible point transformations is defined, called class S. Any measurepreserving point transformation induces a unitary operator on the Hilbert space of 22-functions. A theorem is proved here which implies that the operator induced by any transformation in class S has simple spectrum. [It is then a known result that the transformations in class S have zero entropy.] Let (X, 9, ,u) be a measure space, isomorphic to the unit interval with Lebesgue measure. A measurable, measure-preserving, invertible point transformation of X is called an automorphism of (X, i, u). A class of automorphisms, called class S for brevity, is defined below (Definition (4)). The purpose of this paper is to prove the following theorem: (1) THEOREM. Let r be an automorphism in class S. Then there exist arbitrarily small sets whose characteristic functions each generate ?2(dM) under the action of the unitary operator UT, where Ur is defined by UfQ(rx) =f (x). In particular U7 has simple spectrum. (2) DEFINITION. Let H be a Hilbert space, T a bounded normal operator on H. Let vE H. Let H(v) consist of the closure of the set of all elements of the form P(T, T*)v, where P(T, T*) denotes a polynomial in T and T*. To say that a vector vEH "generates H under the action of T" means that H= H(v). (3) DEFINITION. Let t = {IAt 1 < i ? m } be a finite, ordered collection of mutually disjoint measurable sets. Then t is called a partition. The union of the members of t need not be the whole space. Let ik be a sequence of partitions with the property that for every measurable set E, there exists a sequence of sets Ek such that each Ek is a union of members of (k, and pA(E A Ek)-*O as k-oo. Then it will be said that ik--e. Here e denotes the partition of the whole space into one-point sets. (4) DEFINITION. Let r be an automorphism of (X, iY, IA), {={A i1Ii

Classifying almost-disjoint families with applications toβN − N

Abstract: A family of infinite subsets of the setN of natural numbers will be called almost disjoint iff any two of its members have finite intersection. We shall define such a family ℱ to ben-separable iff for every decompositionD = {D 1, …D n } ofN inton or fewer disjoint subsets there exist setsF ∈ ℱ andD ∃D such thatF ⊆D, and we shall use this and related notions to classify almost-disjoint families, using, on occasion, special axioms of set theory.

A compact set of disjoint line segments in E 3 whose end set has positive measure

Abstract: If L is a set of disjoint closed line segments in E n , let E ( L ) denote the end set of L , i.e. the set of end points of members of L . In [ 1, 2 ] V. L. Klee and M. Martin proved the following lemma: If L is a disjoint set of closed line segments in E 2 such that E ( L ) is compact, then E ( L ) has zero 2-measure.

Optimum Network Partitioning

Abstract: The problem of finding the optimum partition of a set of objects into disjoint collectively exhaustive subsets has been treated by several authors as a special set-covering problem. This paper considers this problem when the objects can be represented as nodes in a directed acyclical network and the subsets of the partition define subnetworks of the original network. A dynamic-programming algorithm is presented that solves the latter problem. This algorithm is, in some important cases, more efficient than the implicit-enumeration schemes that have been previously proposed for the more general set-partitioning problem.

A Minimal Pair of $\Pi^0_1$ Classes

Abstract: A pair of sets (A0, A1) forms a minimal pair if A0 and A1 are nonrecursive, and if whenever a set B is recursive both in A0 and in A1 then B is recursive. C. E. M. Yates [8] and independently A. H. Lachlan [4] proved the existence of a minima] pair of recursively enumerable (r.e.) sets thereby establishing a conjecture of G. E. Sacks [6]. We simplify Lachlan's construction, and then generalize this result by constructing two disjoint pairs of r.e. sets (A0, B0) and (A1B1) such that if C0 separates (A0, A1 and C1 separates (B0, B1), then C0 and C1 form a minimal pair. (We say that C separates (A0, A1) if A0 ⊂ C and C ∩ = ∅.) The question arose in our study of (Turing) degrees of members of certain classes, where we proved the weaker result [2, Theorem 4.1] that the above pairs may be chosen so that C0 and C2 are merely Turing incomparable. (There we used a variation of the weaker result to improve a result of Scott and Tennenbaum that no complete extension of Peano arithmetic has minimal degree.)

Continuum neighborhoods and filterbases

Abstract: Introduction. In this paper S denotes a compact Hausdorff space. If pES and WCS, then W is a continuum neighborhood of p iff W is a subcontinuum of S and&pEInt(W). If ACS, T(A) denotes the complement of the set of those points p of S for which there exists a continuum neighborhood which is disjoint from A [1]. S is said to be T-additive iff for every collection A of closed subsets of S whose union is closed, T(UA) = U I T(L) I L EA } [2 ]. The following three theorems are established.

Harmonic Analysis on Totally Disconnected Sets

Abstract: Preliminaries from fourier analysis and integration theory.- Pseudo-measures supported by totally disconnected sets.- A characterization of uniqueness sets.- Independent sets and arithmetic progressions.- Kronecker's theorem and Kronecker sets.- Independent sets of multiplicity.- Helson sets.- Concluding remarks.- The Wiener process.- Malliavin's theorem.

A generalization of the divide-sort-merge strategy for sorting networks

Abstract: With a few notable exceptions the best sorting networks known have employed a "divide-sort-merge" strategy. That is, the N inputs are divided into 2 groups - - normally of size $\lceil \frac{1}{2} N\rceil$ and $\lfloor \frac{1}{2} N\rfloor$ [Here $\lceil x\rceil$ denotes the smallest integer greater than or equal to x, whereas $\lfloor x\rfloor$ denotes the largest integer less than or equal to x] - - that are sorted independently and then "merged" together to form a single sorted sequence. An N-sorter network that uses this strategy consists of 2 smaller sorting networks followed by a merge network. The best merge networks known are also constructed recursively, using 2 smaller merge networks followed by a simple arrangement of $\lceil \frac{1}{2} N\rceil$ - 1 comparators. We consider a generalization of the divide-sort-merge strategy in which the N inputs are divided into g $\geq$ 2 disjoint groups that are sorted independently and then merged together. The merge network that combines these g sorted groups uses d $\geq$ 2 smaller merge networks as an initial subnetwork. The two parameters g and d together define what we call a "[g,d]" strategy. A [g,d] N-sorter network consists of g smaller sorting networks followed by a [g,d] merge network. The initial portion of the [g,d] merge network consists of d smaller merge networks; the final portion, which we call the "f-network," includes whatever additional comparators are required to complete the merge. When g = d = 2, the f-network is a simple arrangement of $\lceil \frac{1}{2} N\rceil$ - 1 comparators; however, for larger g,d the structure of the [g,d] f-network becomes increasingly complicated. In this paper we describe how to construct [g,d] f-networks for arbitrary g,d. For N < 8 the resulting [g,d] N-sorter networks are more economical than any previous networks that use the divide-sort-merge strategy; for N < 34 the resulting networks are more economical than previous networks of any construction. The [4,4] N-sorter network described in this paper requires $\frac{1}{4} N{(log_2 N)}^2\ - \frac{1}{3} N(log_2 N) + O(N)$ comparators, which represents an asymptotic improvement of $\frac{1}{12} N(log_2 N)$ comparators over the best previous N-sorter. We indicate that special constructions (not described in this paper) have been found for [$2^r , 2^r$] f-networks, which lead to an N-sorter network that requires only .25 $N{(log_2 N)}^2\ - .372 N(log_2 N) + O(N)$ comparators.

Locatability of Faults in Combinational Networks

Abstract: A formal model for the study of reliable combinational networks is introduced and used to determine network properties conducive to the location of faults. The usual concept of fault location is generalized to be an interval on the partially ordered set of subsets of network nodes that classifies nodes into three disjoint sets: a faulty set, a fault-free set, and an indeterminate set. After developing basic properties of locatable faults, necessary and sufficient conditions for a fault to be locatable in an arbitrary network are stated and proven.

Summing closed $U$-sets for Walsh series

Abstract: The union of countably many closed sets of uniqueness for the Walsh series is again a set of uniqueness In a classic paper on Walsh series [1] Sne1der proved the finite union of closed sets of uniqueness, Es, for the Walsh series is again a set of uniqueness In case there were countably many sets of uniqueness he needed to further assume that EiC Vi, where V1, * * *, Vr, * * *, formed a disjoint collection of open intervals Combining recent developments in Walsh series [2 ] with a slight generalization of a classical lemma (see Lemma 3) we will show this further assumption is unnecessary The proof is similar to the proof of the trigonometric analogue given in [3 ] We briefly review the definition of the Walsh functions: let o0(x) -1 if 0 0, then the (n+l)th Walsh fuinction is In(x) =0n,(x) * * * on,(x) where n= = 2ni uniquely determines the ni by specifying ni+i 1 ak'k(X) is a Walsh series such that Received by the editors July 31, 1970 AMS 1969 subject classifications Primary 4215; Secondary 3340

Sums of sets of continued fractions

Abstract: For each integer k jjj 2, let S(k) denote the set of real numbers a such that OaaaA-1 and a has a continued fraction containing no partial quotient less than k. It is proved that every number in the interval [0, 1 ] is representable as a sum of k elements of S(k). For each integer k}±2, let S(k) denote the set of real numbers a such that O^a^k"1 and a has a continued fraction containing no partial quotient less than k (here 0 is to be regarded as the reciprocal of an infinite partial quotient, so 0 belongs to S(k)). Define the sum A +B of two point sets A and B to be the set of all a + b, where a is in A and b is in B. Define the sum ^4+^4+ • ■ • +A (n summands) inductively for each integer w^2, and let nA denote the resulting point set. Theorem 1 of paper [l] by the first author is equivalent to the assertion that 5(2)+ 5(2) = [0, l]. In this paper we prove the following generalization of this result: Theorem. For each integer k^2, kS(k) = [0, l]. We make use of the fact that for each integer & = 2 the set S(k) may be obtained from [0, krl ] by the removal of a suitable infinite set of disjoint open intervals. In fact, as explained in [l], S(k) belongs to the class of Cantor point sets, which are defined by the following procedure: Take a closed interval A = [x, x+a] on the real line, and remove from it a middle open interval Ai2 = (x+ai, x+ai+ai2); two closed intervals Ai = [x, x+ai] and A2— [x+ai+ai2, x+a] remain. The second stage of the procedure is the removal of middle open intervals from Ai and A2. The process is continued, so the wth stage of the procedure is the removal of 2n_1 middle open intervals. The set which results in the limit, when every closed interval which arises has been subdivided, is a Cantor point set. To begin the procedure for obtaining S(k) as a Cantor point set, we take A = [0, k-1]. Let CF(0, au a2, ■ ■ ■ ) denote the continued fraction with partial quotients 0, alt a2, • ■ • . In defining the subPresented to the Society, April 9, 1971; received by the editors November 2, 1970. A MS 1970 subject classifications. Primary 10F20, 10J99; Secondary 10K15.

Stone-Weierstrass theorems for $C(X)$ with the sequential topology

Abstract: The spaces P with one of the following two properties are studied: every continuous function is a Baire function with respect to any algebra a of continuous functions such that (a projectively generates the topology, or with respect to any algebra which distinguishes the points. The former property is equivalent to the statement that of any pair of disjoint zero sets at least one is Lindelof, the latter implies that the space is Lindelof and is implied by analyticity. Connections with the Blackwell problem are shown. The main results are Theorems 2 and 3 below. Note that by a function we always mean a real valued function, and by an algebra of functions we mean an algebra of functions (with the usual "pointwise" operations) which contains all constant functions. Unless explicitly stated otherwise, by a space we mean a completely regular topological space. In general we use the terminology and notation of [1]. In ?1, the results will be stated, and the proofs of results called theorems will be given in ?2. 1. The collection Baire(P) of all Baire sets in a topological space P is the smallest a-algebra of subsets of P such that all continuous functions are measurable. Recall that Baire(P) is the smallest countably additive and countably multiplicative collection of sets, which contains all zero sets (equivalently, cozero sets) in P. A space P locally belongs to a collection 5M? of subsets of P if each point of P has arbitrarily small neighborhoods in on. We are prepared to state our first result. THEOREM 1. A space P is Lindeldf if and only if the collection of all Baire sets in P is the smallest countably additive and countably multiplicative collection J of sets such that P locally belongs to S. In order to state the next result we need more notation and terminology. Denote by F(P) the algebra of all functions on P. For each MC F(P) let uM be the set of all pointwise limits of sequences in M. Received by the editors March 21, 1970. AMS 1969 subject classifications. Primary 2810, 5440, 2635; Secondary 4625.

Truth table classification and identification.

Abstract: A logical basis for classification is that elements grouped together and higher categories of elements should have a high degree of similarity with the provision that all groups and categories be disjoint to some degree. A methodology has been developed for constructingclassifications automatically that gives nearly instantaneous correlations of character patterns of orgnisms with time and clusters with apparent similarity. This means that automatic numericalidentification will always construct schemes from which disjoint answers can be obtained if test sensitivities for characters are correct. Unidentified organisms are recycled through continuous classification with reconstruction of identification schemes. This process is cyclic and self-correcting. The method also accumulates and analyzes data which updates and presents a more accurate biological picture.

Über die Kreuzungszahl vollständiger, n-geteilter Graphen

Abstract: Let K(x1, x2, , xn) be a graph without loops or multiple edges, the complement of which consists of n disjoint complete graphs of x1, x2, , xn vertices. In this paper a class of mappings of K(x1, , xn) onto the Euclidean plane is described. The minimum number of intersection points of edges for these mappings is determined. This number also involves an upper bound for the so-called crossing number cr(x1, , xn), being the minimum number of intersection points of edges for all mappings of K(x1, , xn) onto the Euclidean plane (see (28)). Equality in (28) is conjectured.

A Characterization of SH-Sets

Abstract: Let G be a locally compact abelian group, and 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). 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 Ko and K1 of K, we can find a function u in A(G) such that ||ull 2, then there is a positive constant a with the following property: (.Ca) For any disjoint closed subsets Ko and K1 of K, there is a character y in G such that inf{ I Iy(xo) 'y(xi) |x CKj, j = 0, 1} > a. Using an analogous argument as in [2, Lemma 7], one can easily prove that every compact set K with property (XCa) satisfies the following condition for some positive constant b: (XCb) For any disjoint closed subsets Ko and K1 of K, there is a function u in A (G) such that ||U||A(G)

Chains of quadratic residues

Abstract: The problem considered in this paper is that of finding longest chains of the type: r1, r2, r3, *. , rm, for which the m(m + 1)/2 sums ri + ri+i + r,+2 + * * * + r,, I 5 i g j g m, will be distinct quadratic residues of a given prime p. 1. Let p be a fixed prime and R the set of its (p 1)/2 quadratic residues. In what follows, r's are elements of R and all sums are reduced modulo p. The problem is to find longest chains of the type: (1) ri, r2, r3, * , for which the m(m + 1)/2 sums: (2) Irk : _ iC jS9 m, k-i will all be distinct quadratic residues of p. Trivially, (3) (2m + 1)2 < (4p -3). Since for (1) to be a quadratic residue chain, it is necessary and sufficient that (4) rrl, rr2,rr3, * , rr, be a quadratic residue chain, we can take r, 1 in (1) and this we shall do. 2. A Configuration. Write the quadratic residues of p round the circumference of a circle in order of magnitude. Join ri and r; if ri + ri is in R. It is well known that for any ri, according to whether p is or is not of the form 8t i 1, there are exactly [(p 7)/4] or [(p 3)/4] quadratic residues r of p, for which ri +r R, ri 5 r. Hence, each of the residues gets joined to exactly the same number of quadratic residues as any other. The configuration obtained appears to be of interest. For p = 19, we get Figure 1. Here each point gets joined to four of the remaining eight. It might be noted that for p = 13, the configuration consists of two disjoint triangles and one cannot travel from one point to every other by moving along the straight lines joining the points. This situation does not seem to arise for any other value of p. Received April 16, 1970, revised August 17, 1970. AMS 1970 subject classifications. Primary 10-04, 1OA15.

Analytic-function bases for multiply-connected regions

Abstract: Let E be a nonempty (not necessarily bounded) region of finite connectivity, whose boundary consists of a finite number of nonintersecting analytic Jordan curves. Work of J. L. Walsh is utilized to construct an absolute basis (Qn5 n = O, + 1, +?2, ) of rational functions for the space H(E) of functions analytic on E, with the topology of compact convergence; or the space H(Cl (E)) of functions analytic on Cl (E)=the closure of E, with an inductive limit topology. It is shown that o Qn(z)Q-n-l(w)=l/(w-z), the convergence being uniform for z and w on suitable subsets of the plane. A sequence (Pn, n = 0, + 1, +2 . ) of elements of H(E) (resp. H(Cl (E))) is said to be absolutely effective on E (resp. Cl (E)) if it is an absolute basis for H(E) (resp. H(Cl (E))) and the coefficients arise by matrix multiplication from the expansion of (Qn). Conditions for absolute effectivity are derived from W. F. Newns' generalization of work of J. M. Whittaker and B. Cannon. Moreover, if (Pn, n = 0, 1, 2, . . .) is absolutely effective on a certain simply-connected set associated with E, the sequence is extended to an absolutely effective basis (Pn, n = 0, + 1, + 2, . . .) for H(E) (or H(Cl (E))) such that En=o P,(Z)Pn 1(w)= 1/(w-z). This last construction applies to a large class of orthogonal polynomials. 1. Interpolation bases. For any subset S of the extended complex plane, let H(S) be the set of functions analytic on S (that is, f is in H(S) if and only if f is analytic on some open set containing S), and zero at infinity if the point at infinity is in S. The convergence of a sequence of elements (f,) of H(S) will be said to be compact-open on S if and only if the sequence converges uniformly on compact subsets of some open set containing S. Throughout the paper, let E be a nonempty region of finite connectivity, whose boundary consists of a finite number of nonintersecting analytic Jordan curves. If E is bounded, divide the boundary curves into two mutually disjoint sets, let L(O) and L(1) be the respective unions of the curves in each set, and suppose that the curve exterior to all the others is a subset of L(1). Let F be harmonic in E and continuous on the closure of E, and take on the values 0 and 1 on L(O) and L(1) respectively. Then there exist points z1, Z2, Z3,. . ., in the components of the complement of E bounded by L(O), and points W1, W2, W3, .. ., in the components of Received by the editors October 7, 1969 and, in revised form, April 17, 1970. AMS 1969 subject classifications. Primary 3070, 3030.

A pointwise convergence theorem for sequences of continuous functions.

Abstract: Let {fk} be a sequence of continuous real valued functions defined on an interval I and N a fixed nonnegative integer such that if fk(x) =fi(x) for more than N distinct values of x E I then fk(X) -fi(x) for x E I. It follows that there is a subsequence {gj} of {fk} such that for each x the subsequence {gj(x)} is eventually monotone. Thus lim1 -. +00 gj(x) =f(x) exists for all x, where f is an extended real valued function. If Ifk(x)I is bounded for each x E I then limj 1 + X gj(x) =f(x) exists as a finite limit for all x E I. For N= 0 this reduces to picking a monotone subsequence from a sequence of continuous functions whose graphs are pairwise disjoint.

Two cells with points of local nonconvexivity

Abstract: A subset S of the plane is a two cell provided S is homeomorphic to { x I I I xI I _ I} . THEOREM. Let S be a two cell with exactly n points of local nonconvexity. Then S is expressible as a union of n+1 compact convex sets with mutually disjoint interiors.

Some Results on Invariant Sets for Translation Parameter Family of Probability Measures--I

Abstract: 1. Preliminaries. Let ,u be a given probability measure on (X, B), where X is some finite dimensional Euclidean space and B is the class of Borel sets on X. For each 0 E X, let Il6(A) = p(A-0), where A C B and A-0 = {x: x+0 c A}. A set 'A' is called ,-invariant if ilo(A) = p(A) V 0 E X. For example the null set 0 and the whole space X are trivially p-invariant. The class of all p-invariant sets is denoted by A(,u). A set 'A' is called "non-trivial" if 0 < ,l(A) < 1. That "nontrivial" p-invariant sets exist is seen by noting that if ,I assigns probability I each to {O} and {1}, then A is ,u-invariant if and only if A' = A+1, e.g., A = -O (2n, 2n+ 1] is p-invariant with p(A) = 2. It is easily seen that A(,) is a monotone class, is closed for complementation and disjoint unions, and is not necessarily a r-algebra. The probability measure ,u is called weakly incomplete (weaklycomplete) if it has (or does not have) non-trivial It-invariant sets. The results we present here have originated from a paper of Basu and Ghosh (1969) on p-invariant sets. Our main object is to make a careful study of some of the conjectures contained in their paper. A brief account of the results contained in the paper is as follows:

A splitting theorem for simple Π 1 1 Sets

Abstract: As was first mentioned in [3, §5], if A is any – set, A is the union of two disjoint – sets B (0), B (1). In metarecursion theory this is proven as follows. Let ƒ be a one-to-one metarecursive function whose range is A , let R be an unbounded metarecursive set whose complement is also unbounded, and set B(0) = f(R), B(1) = f ( ). The corresponding fact of ordinary recursion theory, namely that any r.e. but not recursive set can be split into two other such sets, was proved by Friedberg [2, Theorem 1], using a clever priority argument. Sacks [7, Corollary 2] then showed that any r.e. but not recursive set is the union of two disjoint r.e. sets neither of which was recursive in the other, a much stronger result. In this paper we attempt to prove the analogous result for – sets A, but succeed only in the case A is simple ; i.e., the complement of A contains no infinite subset. As a corollary we show the metadegrees are dense, a fact already announced by Sacks [8, Corollary 1], but only proven by him for nonzero metadegrees.

An elementary proof of Abramov's result on the entropy of a flow

Abstract: Let T be an automorphism on a probability space (Ω, B, P) Given a finite partition α of Ω , into disjoint measurable sets A 1 , A 2 , · · ·, A n , its entropy is (D1)

A certain discontinuous markov process in stellar dynamics

Abstract: The basic assumption in Chandrasekhar’s approach of statistical stellar dynamics (Chandrasekhar, 1942) is the postulate that a test star within a stellar system being stationary in the sense of collisionless continuum theory suffers random displacements in velocity space generated by the fluctuating part of the gravitational field in a manner that can be described in terms of a random walk. This is equivalent to the assertion that the increments of velocity are regarded as stochastically independent in disjoint time intervals. From this Chandrasekhar derived a diffusion process in velocity space. The equation of motion of the probability density W(r, u, t) in the whole 6-dimensional phase space is then written in the form of a Fokker-Planck-type equation: $$\frac{{\partial W}}{{\partial t}}+u\cdot{ abla_r}W+{ abla_r}\Phi\cdot{ abla_u}W={ abla_u}\left({q{ abla_u}W+\eta{W_u}}\right)$$ (1) r, u = position, velocity vector; Φ = gravitational potential of the ‘smoothed out’ distribution of matter; q = diffusion coefficient; and η = coefficient of dynamical friction.

Pushing apart disjoint closed sets in normed linear spaces

Abstract: It is well known that disjoint closed sets in a Euclidean space may be pushed apart a positive distance by a space homeomorphism. Answering a question raised by R.D. Anderson and F.E. Browder, we show that the same thing can be accomplished in any normed linear space with respect to any locally finite collection of disjoint closed sets. The desired space homeomorphism is obtained by pushing outward along rays from a point not contained in the union of the closed sets. An example showing this result does not hold for the non-normable Frechet spaces.