Chainable Subcontinua

This paper is concerned with conditions under which a metric continuum (a compact connected metric space) contains a nondegenerate chainable continuum. This paper is concerned with conditions under which a metric continuum (a compact connected metric space) contains a non-degenerate chainable continuum. By R.H. Bing’s theorem eleven [2] if a metric continuum X contains a non-degenerate subcontinuum H which is hereditarily decomposable, hereditarily unicoherent, and atriodic, then H is chainable. The following papers give examples of continua with the property that each non-degenerate subcontinuum is not chainable. G.T. Whyburn [16]. R.D. Anderson and G. Choquet [1]. A. Lelek [6] gives an example of a planar weakly chainable continuum each non-degenerate subcontinuum of which separates the plane and thus contains no non-degenerate chainable subcontinuum. W.T. Ingram [5] gives an example of an hereditarily indecomposable tree-like continuum such that each non-degenerate subcontinuum has positive span and hence is not chainable. C.E. Burgess in [3] shows if a continuum M is almost chainable and K is a proper subcontinuum of M which contains an endpoint p of M , then K is linearly chainable with p as an end point. A continuum M is almost chainable if, for every positive number ε, there exists an ε-covering G of M and a linear chain C(L1, L2, . . . , Ln) of elements of G such that no Li (1 ≤ i < n) intersects an element of G − C and every point of M is within a distance ε of some element of C. He also shows if M is almost chainable, then M is not a triod and M is unicoherent and irreducible between some two points. Examples show M can contain a triod or a non-unicoherent subcontinuum. If X and Y are metric continua and if X can be ε-mapped onto Y for all positive ε and Y has a non-degenerate chainable continuum then so does 2000 Mathematics Subject Classification. 54F20.