ALMOST EVERYTHING YOU WANTED TO K'NOW ABOUT HOMOGENEOUS, CIRCLE-LIKE

A continuum is a compact, connected, nonvoid metric space. A continuum X is homogeneous if for each pair of points x and y in X, there exists a homeomorphism of X onto itself that maps x to y. The simple closed curve is the only obvious homogeneous plane continuum. In an effort to construct other homogeneous plane con tinua, one might consider the chainable continuum pictured in Figure 1 and formed from the product of the Cantor set and the interval [0,1] by adding horizontal line segments. The first horizontal line segment joins (1/3,0) and (2/3,,0); the second and third join the points (1/9,1) and (2/9,1) and the points (7/9,1) and (8/9,1), respectively. The horizontal line segments continue to alternate between the top and bottom according to the length of the gaps of the Cantor set. Call this continuum Xl