Completely regular mappings and homogeneous, aposyndetic continua

James T. Rogers
1981 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
The purpose of this note is to prove an improved version of Jones' Aposyndetic Decomposition Theorem. Corollaries to the new theorem re-emphasize the importance of understanding aposyndetic, homogeneous continua. The proof is a synthesis of results about homogeneous continua with results from an unexpected source: completely regular mappings. Completely regular mappings occur naturally and often in the study of homogeneous continua, which is a surprising and pleasing phenomenon, since these
more » ... ings were invented for quite another purpose [1] . The author believes that these maps are likely to provide even more new information about homogeneous continua. A continuum is a compact, connected, nonvoid metric space. A curve is a one-dimensional continuum. A continuum M is homogeneous if for each pair of points p and q belonging to M, there exists a homeomorphism h:M-> M such that h(p) = q. A mapping f:X -• » F of X onto F is completely regular [1] if given e > 0 and y £ F, there exists an open set V in Y containing y such that if y' £ V, then there is a homeomorphism h îromf~l (y) to/ -1 (y') such that d (x, h(x)) < e. Each completely regular mapping is open. We will need the following theorem, which should be well-known. A proof is included for completeness. THEOREM 1. If X is a curve and H 1 (X) = 0, then X is hereditarily unicoherent.
doi:10.4153/cjm-1981-039-4 fatcat:oupv4tzjpbc67ht5p5vauc6ypa