Each Homogeneous Nondegenerate Chainable Continuum Is a Coset Space

Let A" be a compact Hausdorff homogeneous space and G be the full group of homeomorphisms from X to X. We denote the isotropy group at x by Gx. If G is topologized by uniform convergence [l], then it is well known that G is a topological group, and the function 6: G/GX—*X, defined by 6{gGx)=g{x) is one-one and continuous. It is an open problem whether ô is a homeomorphism, i.e., whether X is a coset space. In this note we prove that each homogeneous nondegenerate chainable continuum M is such a space. Ford [2]2 has proved that every SLH (strongly locally homogeneous) completely regular Hausdorff space is a coset space. We will prove that M is not an SLH space but still a coset space. Bing [3] has proved that each homogeneous nondegenerate chainable space is a pseudo-arc. Therefore we only need to prove that the pseudo-arc is a coset space. In his papers [3; 4; 5], Bing gave a very beautiful treatment of the pseudo-arc. We will follow his definitions and notations, and we will not repeat all his definitions here. Description of the pseudo-arc M [4, p. 730]: Let P and Q be two points of a compact metric space and D\, D2, • • • a sequence of chains from P to Q such that for each positive integer i, (1) Di+i is crooked in Dit (2) no link of 7?¿ has a diameter of more than 1/2' and (3) the closure of each link of Di+i is a subset of a link of 7>,. Then M is the common part of D*, D*, • • • , where D* denotes the sum of the links of 7\. It is well known that the pseudo-arc is an indecomposable continuum [4]. Definition. A space X is an SLH space if for any x£ X, and neighborhood Vof x, there is a neighborhood Uof x, UGV, such that for any y£ V, there is a homeomorphism g of A such that g(x) =y and g(z)=z if zGU.