Terminal continua and quasi-monotone mappings (Q1207242)

In their monograph [Diss. Math. 149 (1977; Zbl 0354.54017)] on nonseparating subcontinua the reviewer and \textit{J. B. Fugate} defined and considered terminal and end subcontinua. A proper subcontinuum \(K\) of a continuum \(X\) is a terminal continuum of \(X\) if whenever \(A\) and \(B\) are proper subcontinua of \(X\) having union equal to \(X\) and \(A\cap K\neq\emptyset\neq B\cap K\), then either \(X=A\cup K\) or \(X=B\cup K\). In this paper this definition is used and should not be confused with ``terminal continua'' as defined by various other authors. A proper subcontinuum \(K\) of a continuum \(X\) is an end continuum of \(X\) provided \(X\) is not the union of two proper subcontinua each intersecting \(K\). In the paper mentioned above, the authors established that a subcontinuum \(K\) is an end continuum of a continuum \(X\) if and only if \(K\) is a terminal continuum with empty interior. In this paper the author shows that if there is a continuous, subjective, quasi-monotone function between continua, then the image of a terminal (an end) subcontinuum of the domain is either a terminal (an end) subcontinuum of the range or equals the entire range. Some other results are also obtained.