On continuously type \(A^{\prime} \theta\)-continua (Q2788871)

\textit{R. W. Fitzgerald} [in: Stud. Topol., Proc. Conf. Charlotte, N. C., 1974, 139--173 (1975; Zbl 0306.54044)] showed that a continuum \(X\) is a \(\theta\)-continuum (\(\theta_{n}\)-continuum, for some positive integer \(n\)) if for each subcontinuum \(K\) of \(X\), we have that \(X\setminus K\) only has a finite number of components (\(X\setminus K\) has at most \(n\) components).NEWLINENEWLINELet \(X\) be a \(\theta\)-continuum type of \(A\). In the paper under review, the author shows that \(X\) is a continuously type \(A^{\prime}\) \(\theta\)-continuum if and only if \(\;T_{X}\) is continuous. Moreover, the author shows that \(G=\{T(\{x\})\mid x\in X \}\) is the finest monotone continuous decomposition of \(X\) such that \(X\setminus G\) is a finite graph. Then as a corollary, the author expresses that if \(X\) is a continuously type \(A^{\prime}\) \(\theta\)-continuum, that is not a finite graph, then \(X\) is not hereditarily decomposable.NEWLINENEWLINEA map \(f:X\twoheadrightarrow Y\) between continua is \textit{atomic} provided that for each subcontinuum \(K\) of \(X\) such that \(f(K)\) is nondegenerate, \(K=f^{-1}(f(K))\).NEWLINENEWLINELet \(X\) be a continuously type \(A^{\prime}\) \(\theta\)-continuum, that is not a finite graph. The author proves that the quotient map is atomic by showing that if \(q: X\twoheadrightarrow D\) is the quotient map, where \(D\) is a finite graph, then \(q\) is atomic.NEWLINENEWLINEThe author gives a partial answer to a question of R. W. Fitzgerald [loc. cit.] which was known to have a negative answer by [\textit{J. Heath}, Houston J. Math. 9, 477--487 (1983; Zbl 0529.54031)] with the following corollary ``If \(X\) is a continuously type \(A^{\prime}\) \(\theta\)-continuum, then there exists a positive integer \(n\) such that \(X\) is a \(\theta_{n}\)-continuum''.NEWLINENEWLINEMoreover, the author proves that ``If \(X\) is a continuously type \(A^{\prime}\) \(\theta\)-continuum such that for each \(x\in X\), \(T_{X}(\{x\})\) is nondegenerate, then \(F_{n}(X)\) is a \(Z\)-set in both \(2^{X}\) and \(C_{n}(X)\) for every positive integer \(n\).''