R. H. Bing

The Geometric Topology of 3-Manifolds

Planar complexes PL planar maps The Schoenflies theorem Wild 2-spheres The generalized Schoenflies theorem The fundamental group Mapping onto spheres Linking Separation Pulling back feelers Intersections of surfaces with

Partitioning continuous curves

1

A connected countable Hausdorff space

-simplexes Intersections of surfaces with skeleta Side approximation theorem The PL Schoenflies theory for

The Kline sphere characterization problem

R^3

A convex metric for a locally connected continuum

Covering spaces Dehns lemma Loop theorem Related results AppendiX: Some standard results in topology References Index.

A convex metric with unique segments

1. The continuous curve. By a continuous curve we mean a compact, locally connected, metric continuum. For definitions of locally connected, continuum, and other terms used in this paper, see either [18] or [2 l ] . Those who like to visualize topology concretely may wish to think of a continuous curve as a chunk out of Euclidean 3-space—one that is connected (all in one piece), one that is bounded (lies on the interior of a sphere), and one that is locally connected (nearby points belong to small connected subsets). A wad of paper, an irregularly shaped rock, or the earth itself may be considered as examples. However, our remarks about continuous curves will apply equally well to those in Euclidean spaces of all dimensions and to those in a Hubert space. When Jordan first introduced the term continuous curve, he defined it analytically to be the image (in the plane) of a straight line interval under a continuous transformation. I t was not until over twenty years later that it was discovered that any compact locally connected metric continuum was the image of a straight line interval under a continuous transformation and conversely. This interesting and unusual discovery adds spice to the study of mathematics [24, p. 12]. Another interesting aspect of this discovery is that it was made independently by two mathematicians, Hahn and Mazurkiewicz. Since Peano had shown earlier that a square plus its interior is the image of a straight line interval, a continuous curve is sometimes called a Peano continuum. In this discussion we shall be interested in the continuous curve itself and not the continuous transformation of an interval. Hence, we use the definition in the first paragraph rather than the analytic one. In this discussion we shall be interested in the structure of a continuous curve.

Retractions Onto Anr's

In Math. Ann. vol. 94 (1925) pp. 262-295, Urysohn gave an example of a connected Hausdorff space with only countably many points. Here is another. EXAMPLE 1. The points of the space are the rational points in the plane on or above the x-axis. If (a, b) is such a point and e>0, (a, b)+{(r, 0)1 either 1r-(a+b/31I2)| <e or |r-(a-b/31I2)| <4l is a neighborhood. To construct geometrically a neighborhood with center at (a, b), consider an equilateral triangle with base on the x-axis and apex at (a, b). (If b=0, regard (a, b) as the triangle.) Then (a, b) plus all rational points on the x-axis whose distances from a base vertex of the triangle are less than e is an e-neighborhood with center at (a, b). This space satisfies the Hausdorff axioms and has the property that for each pair of neighborhoods, there is a point common to their closures. Hence, the space is connected. Although this space has a countable basis, it is not regular and hence not metric. Its dimension depends on the definition of dimension used. In the Menger-Urysohn sense (the dimension is defined inductively in terms of boundaries of open sets) the space is one-dimensional, and in the Lebesgue sense (the dimension is defined in terms of orders of coverings) it is infinite-dimensional. EXAMPLE 2. We may enlarge our description to get a connected countable Hausdorff space of any positive dimension (even infinite) in the Menger-Urysohn sense. We give one of dimension two. The points of the space are the rational points of Euclidean 3-space whose second and third coordinates are non-negative. If (a, b, 0) is a point, an e-neighborhood with center at (a, b, 0) is (a, b, 0) + { (r, 0, 0) 1 either I r-(a+b/31I2) 1 <e or I r-(a-b/31I2) 1 <e} . If (a, b, c), c

A DECOMPOSITION OF S3 WITH A NULL SEQUENCE OF CELLULAR ARCS

0, is a point, an e-neighborhood with center at (a, b, c) is the sum of (a, b, c) and all e-neighborhoods with centers at points (a, r, 0) where either I r-(b+c/21I2) 1 <e or I r-(b-c/21I2)1 <e.

Concerning simple plane webs

The object of this paper is to give a solution to the following problem proposed by J. R. Kline: Is a nondegenerate, locally connected, compact continuum which is separated by each of its simple closed curves but by no pair of its points homeomorphic with the surface of a sphere? The answer is in the affirmative. A solution to the Kline problem gives a characterization of a simple closed surface. Partial solutions of this problem have been made by Hall [l, 2 ] 1 and Jones [3]. Other characterizations of a simple closed surface have been given by Kuratowski [4], Zippin [5, 6] , Wilder [7] and Clay tor [8]. Previous to the giving of these characterizations, Moore gave [9] two sets of axioms, each set of which characterized a set topologically equivalent to a plane. DEFINITION. We say that M disrupts X from Y in D if there is an arc from X to Y in D but each such arc contains a point of M. We shall make use of the following lemma.

Metrization of topological spaces

A subset M of a topological space 5 is said to have a convex metric (even though S may have no metric) if the subspace M of 5 has a convex metric. It is known [5 J that a compact continuum is locally connected if it has a convex metric. The question has been raised [5] as to whether or not a compact locally connected continuum M can be assigned a convex metric. Menger showed [5] that M is convexifiable if it possesses a metric D such that for each point p of M and each positive number e there is an open subset R of M containing p such that each point of R can be joined in M to p by a rectifiable arc of length (under D) less than e. Kuratowski and Whyburn proved [4] that M has a convex metric if each of its cyclic elements does. Beer considered [ l ] the case where M is one-dimensional. Harrold found [3] M to be convexifiable if it has the additional property of being a plane continuum with only a finite number of complementary domains. We shall show that if M\ and M2 are two intersecting compact continua with convex metrics D\ and D2 respectively, then there is a convex metric D% on M\-\-Mi that preserves D\ on M\ (Theorem 1). Using this result, we show that any compact ^-dimensional locally connected continuum has a convex metric (Theorem 6). We do not