Point Compactification Research Articles

The Birkhoff Center and Analytic Sets

It is shown here how G. D. Birkhoff's notion of the center of a homeomorphism or flow naturally gives rise to an analytic set in a product space. It is shown that for a wide class of spaces this set is not a Borel set. Let X be a locally compact separable metric space with complete metric d and let H(X) be the space of autohomeomorphisms of X. The space H(X) has a topology under which it is a complete separable metric group [6, 9]. For a wide class of Xs, it is known that this topology is unique [7]. This topology may be briefly described as follows. Let X* = X U {oo} be the one point compactification of X and consider the space M = M(X*, X*) of all continuous maps of X* into X* provided with the compact open topology [9]. In this topology, M is a Polish space: M is separable and possesses a complete metric compatible with this topology. Identify H(X) with F = {(f, g) E M x M:fg = gf = idx* and f(oo) = oo}. Since F is closed in M x M, F is also a Polish space. We consider H(X) to have this topology. If h E H(X) and Y is an h-invariant subset of X, then a point y E Y is said to be nonwandering with respect to Y provided there is an increasing sequence of positive integers n1, n2, n3, . . . and points yp E Y, p = 1, 2, 3, . . . such that the sequence hnp(yp) converges to y. Let Rh(Y) = {y E Y:y is nonwandering with respect to Y}. If Y is a closed h-invariant set, then Rh(Y) is also closed and h-invariant. Set R(X) = X and by recursion, for each ordinal ox, Ro '(X) = Rh(Ro'(X)) and, if X is a limit ordinal, Rx(X) = n,a<x R-(X). Since this central sequence {RO'(X)} forms a decreasing transfinite sequence of closed sets in X, there is a least countable ordinal 8 = 8(h) such that Rh'(X) = R'(X). This ordinal is called the depth of h and R'(X) = Rh(X) is called the center of h. Of course, Rh(X) is the closure of the set of all h-recurrent points.

Regularity at infinity for area-minimizing hypersurfaces in hyperbolic space

equipped with the hyperbolic metric y-Z(dxZ+dy2). A standard compactification of IH involves adding the boundary (R"• {0})w {*} so that I[-I is simply the one point compactification of the Euclidean closed half-space R" • [0, ~). Suppose 0 7, any interior singularities of M must remain in a bounded region of hyperbolic space. Near points of F (in the Euclidean topology), M w F may thus be described as the graph of a function. This function is the solution of an interesting partial differential equation that becomes degenerate along the part of the boundary corresponding to F. The second author has recently established [L] a boundary higher-regularity result for this equation, which implies, in particular, that M w F is r if F is cgk,~ for k=2 , 3 .... , ~ . Finally, in case F bounds a star-

The quasihyperbolic metric and associated estimates on the hyperbolic metric

In this paper we investigate the geometry of the quasihyperbolic metric of domains in R". This metric arises from the conformally flat generalized Riemannian metric d(x, aD)-'ldx I. Due to the fact that the density cl(x, aD) -t is not necessarily differentiable, the classical theories of Riemannian geometry do not apply to this metric. The quasihyperbolic metric has been found to have many interesting and varied applications in geometric function theory. In particular, quasiconformal mappings are quasi-isometries of this metric for sufficiently far-lying points, also bounds on the quasihyperbolic metric in terms of other metrics imply that a domain is uniform which then implies certain injectivity criteria for locally-Lipschitz mappings, amongst others. In fact there is quite a strong relationship between uniform domains and the quasihyperbolic metric. Most of these basic results on the quasihyperbolic metric can be found in [3], [2] and [5]. We note here that the quasihyperbolic metric is complete and generates the usual topology on a proper subdomain of R". Further, geodesics (length minimizing curves) always exist for this metric and these geodesics have Lipschitz continuous first derivatives, which is in fact best possible. We begin by calculating the quasihyperbolic metric, its curvature and geodesics for some nontrivial examples in R'. We also calculate the possible isometries and show that they are conformal mappings, and so when n > 2 are M6bius transformations. We then turn to the planar case for a more detailed investigation. Here it becomes possible to compute the Gaussian curvature of the quasihyperbolic metric in some basic examples and use these for comparison in more general cases. In particular, we show that the quasihyperbolic metric of a planar domain is an S-K metric, in the sense of Heins, if and only if the domain is convex. We denote euclidean n-space by R" and its one point compactification by R'.

Continuous Preimages of Spaces with Finite Compactifications

A compactification αX of the space X is called an n -point compactification if the remainder αX — X consists of exactly n points. K. D. Magill [5] showed that if Y has an n-point compactification and if f:X→ f(x) = Y is a compact continuous mapping of the space X onto Y, then X also has an n-point compactification.

Spaces which Cannot be Written as a Countable Disjoint Union of Closed Subsets

It is well known (see [3](1)) that no continuum (i.e. compact, connected, Hausdorff space) can be written as a countable disjoint union of its (nonvoid) closed subsets. This result can be generalized in two ways into the setting of locally compact, connected, Hausdorff spaces. Using the one point compactification of a locally compact, connected, Hausdorff space X one can easily show that X cannot be written as a countable disjoint union of compact subsets. If one makes the further assumption that X is locally connected, then one can show that X cannot be written as a countable disjoint union of closed subsets.(2)

A characterization of realcompact extensions

Let X be a Tychonov space, R be the real numbers and R*= R u {oo} be its one point compactification. As in [1, 8B], for each g E C(X), let g*:/X-R* be its Stone extension and vgX={p cfX:g*(p)$oo}= g* [R]. Let T be any Tychonov space containing X densely and let f: /X--/#T be the continuous map fixing X pointwise. Let CT C C(X) be that family of functions which can be continuously extended to T. In [2, Theorem 2], it was shown that g E CT if and only if for every sequence (Fn)n= of closed sets in R such that nn=l1 F,= 0, it follows that nnl1 CT g9[FnF= 0.

Bitopological local compactness

In a bitopological space ( X, T 1, T 2), T 1 is said to be locally compact with respect to T 2 if for each point x ϵ X there is a T 1 open neighbourhood of x whose T 2 closure is pairwise compact. ( X, T 1, T 2) is pairwise locally compact if T 1 is locally compact with respect to T 2 and T 2 is locally compact with respect to T 1. This paper examines this concept, defines a bitopological analogue of the Alexandroff one point compactification and investigates its properties.

Local triviality of fiberings

We prove that a Hurewicz fibering or a Serre fibering is locally trivial if the total space is a connected separable metric ANR n-gm over a principal ideal domain and the base space is a weakly locally contractible paracompact finite dimensional space, and all fibers are homeomorphic to a space which is a connected 3-manifold with exactly one end and whose one point compactification is a 3-manifold and it has no false 3-cells, in particular a euclidean 3-space.

Meromorphic functions which cluster on the boundary

Let 41(z) be a univalent meromorphic function defined on U, the unit disc, and let S2 denote the Riemann sphere, that is, the one point compactification of the complex plane R2. Let C(t, eit) be the set of all points v in S2 such that there exists a sequence z:R-eit with 41(zn) at t. It is easily seen that C(Q, eit) CO9/'(U), and furthermore, O9/'(U) has at least two points. If B(z) is a finite Blaschke product, andf(z) =41(B(z)), then C(f, et) COf( U). If B(z) is an infinite Blaschke product, this property is destroyed since 4/(0)CC(f, eit), where the zeros of B(z) cluster at eit. The purpose of this paper is to present the following theorem, characterizing those meromorphic functions clustering on the boundary:

A simply connected 3-manifold is 𝑆³ if it is the sum of a solid torus and the complement of a torus knot

It has been shown [6] that any closed, connected, orientable 3manifold can be constructed by removing a finite number, k, of mutually exclusive tame solid tori1 from the 3-sphere, S3, and then sewing them back in some possibly different manner. In particular, any homotopy 3-sphere can be obtained in this manner; thus it gives an approach to the Poincar6 Conjecture. We wish to consider the case k =1. To this end let T be a tame solid torus in SI and let M be a simply connected 3-manifold containing a solid torus S such that there is a homeomorphism h of S3-Int(T) onto M-Int(S). It is known [I ] that M is S' if T is unknotted or if T has the type of a trefoil knot. In [5] it is asserted that M is SI in every case. The proof given in [51 shows only that h must take some longitude of T onto a longitude of S-a fact that is not sufficient to prove this assertion. The purpose of this paper is to provide a proof that M is S3 in the case that T has the type of a torus knot. We first give a canonical description of the torus knot of type (p, q). We consider 5' as the one point compactification of E3 which we represent in cylindrical coordinates (r, 0, z). Let R be the torus R= { (r, 0, z): (r-2)2+z2= 1}. Let p and q be relatively prime positive integers. We assume, for convenience, that p>q; since it is known that the torus knots of types (p, q) and (q, p) are equivalent. For 1 <k?p let X=h(3, 2k2r/p, 0) and yk= (1, 2kir/p, 0). Let ak be an arc on R from xk to yk+4 that lies in the positive z half space and which increases monotonically with 0. (It is to be understood that the subscripts are to be reduced modulo p.) Let bk be the arc on R from Xk+q to yk+q that lies in the negative z half space and the plane O=2(k+q)7r/p. We denote the simple closed curve UL, (akUbk) by Kp,. Figure 1 shows K6,3.

E 4 is the Cartesian Product of a Totally Non-Euclidean Space and E 1

THEOREM 2. The Cartesian product of H and El is homeomorphic to E4; the suspension of G is homeomorphic to I'; and the suspension of H is homeomorphic to S4. We recall that the suspension of a space is its join with a 0-sphere. We will call an n-dimensional space totally non-euclidean if it contains no open subset homeomorphic to En. Theorems 1 and 2 imply, of course, as corollaries that there are involutions (period two homeomorphisms) of I', E4, and S4 having G, H, and H. respectively, as fixed point sets. Intuitively speaking, His constructed by iterating a process of cutting the interiors of a sequence of tetrahedra E3 and pasting in homeomorphs of Bing's decomposition space. If E' is thought of as the interior of a cube, the quotient map for G is equivalent to that of H on the interior of the cube and is one-to-one on the boundary. H is the one point compactification of H. As Bing's example the nondegenerate elements of the decompositions giving rise to these spaces are tame arcs. In fact, by a construction of M. K. Fort, Jr. [5], we might even suppose that each nondegenerate element is an arc which is the union of three line segments. Part I is devoted to proving Theorem 2. In Part II we develop some results which are used to prove Theorem 1. The work Part II is carried out greater generality than necessary for the main results, but offers, as well, a partial solution to a question raised by Bing.

The end point compactification of manifolds

Introduction^ This paper originated in trying to show that the one point compactification of an orientable generalized ^-manifold (n-grn) with cohomology isomorphic to Euclidean n-space was an orientable n-gm. Heretofore, in papers on transformation groups where this was relevant, it was stated as an extra assumption. The solution to this problem is given as a corollary to the main theorems which characterize the orientable (or locally orientable) generalized manifolds whose Freudenthal end point compactification is again an orientable (or locally orientable) generalized ^-manifold (see 4.5 and 4.13). In the first section we give a new characterization of the Freudenthal end point compactification in terms of inverse limits. We show as in Specker [10] that a certain O-dimensional cohomology group measures the extent of this compactification. Higher dimensional analogues of this cohomology group have been used by Conner [4] in proving, e.g., that a simply connected locally Euclidean ^-manifold whose 1 point compactification is a locally Euclidean w-manifold cannot be fibered by a non-trivial compact fiber. He has called these groups the cohomology of the ideal boundary. These groups are further studied and the homology analogue is derived (see § 2). The main Lemma (2.16) is an exact sequence which relates the homology of the ideal boundary with the homology of a given compactification and the local homology groups at infinity. This exact sequence together with our characterization of the Freudenthal compactification gives the main theorems. Applications are given in § 3 to Poincare duality, in § 5 to open 2manifolds, and in § 6 special mappings of manifolds. Throughout this paper X will denote a locally campct, locally connected, connected Hausdorff space. If S is a locally compact Hausdorff space, A(S) will denote the collection of open subsets of S whose closure is compact. When the generic space is not necessarily locally connected, as is usually the case in § 2 and part of § 4, it will be denoted by the letter S.

Conditional probability distributions in the wide sense

Introduction. Doob [3, p. 29],2 has shown that if Q= [U, Al, u] is a probability space (i.e. cL is a o-field of subsets of a set U and u is a countably additive non-negative measure defined on 'U, normalized by the condition u(U) =1) and cto is a o-subfield of cl, and y is an n-dimensional random variable mapping U into an n-dimensional Euclidean space X, then y possesses a conditional distribution in the wide sense relative to 'to. More specifically, he proved the existence of a function, p, of two variables Y and w, where Y ranges over the Baire subsets of X, and w ranges over U, which satisfies the following three conditions: (i) p(Y, w) is, for each fixed w, a probability measure defined on the Baire subsets of X; (ii) p(Y, w) is, for each Y, a function measurable relative to cUo; and (iii) fAp( Y, w)du(w) =u(Afny-'(Y)) for each A IL0o and each Baire subset Y of X. This result of Doob's is known to be false if one lets X be an arbitrary set with a distinguished o-field of measurable subsets. It is our purpose (Theorem 1, below) to show that Doob's result does hold, at least if X is any locally compact, Hausdorff space whose one point compactification is metrizable. The idea behind this generalization is the following observation. If p(Y, w) exists with properties (i), (ii) and (iii) above then by (i) p(Y, w) determines for each fixed w, a unique element P(w) in the dual of C(X) where C(X) is the Banach space of all continuous realvalued functions on X which vanish at infinity. Furthermore from (ii) it follows that (g, P(w)) is for each g in C(X) a function measurable relative to cIo where (g, P(w)) is the value of the linear functional P(w) at the vector g in C(X). And from (iii) it follows that fA(g, P(w))du(w) =fAg(y(w))du(w) for each AEclto and each g in C(X). This last is equivalent to fA(g, P(w))du(w) -fA(g, y(w))du(w) for each A E?lCo and each g in C(X) where for each fixed w in U, y(w) is the probability measure which lives at the one point y(w). Thus P is nothing other than the conditional expectation of y relative to ItLo. Here P and y are Banach space valued random variables. Thus if y possesses a conditional distribution in the wide sense relative to 'Uo