The cyclic extensibility of essential components of the fixed point set (Q1327653)

Let \(X\) be a continuum. If every continuous mapping \(f:X\to X\) has at least one fixed point, \(X\) is said to have the fixed point property (f.p.p.). We investigate the existence of essential components of the fixed point sets and the property f.*p.p., which are defined as follows: a component \(C\) of the fixed point set of \(f\) is called essential, if for any \(\varepsilon>0\) there exists \(\delta>0\) such that every continuous mapping \(f':X\to X\) with \(|f'-f|<\delta\) has a fixed point in the \(\varepsilon\)-neighborhood \(U_\varepsilon(C)\) of \(C\), and otherwise it is called non-essential; and \(X\) has f.*p.p., if \(X\) has f.p.p., and the fixed point set of every continuous mapping \(f:X\to X\) has at least one essential component [\textit{S. Kinoshita}, Osaka Math. J. 4, 19-22 (1952; Zbl 0047.16204); the author, Math. Jap. 38, 235-238 (1993; Zbl 0808.47045)]. Note that there exists a space which has f.p.p., but does not have f.*p.p. [\textit{Y. Yonezawa}, Fundam. Math. 139, 91-98 (1991; Zbl 0754.54031)]. The Hilbert cube \(I^\omega\) has f.*p.p., and the property f.*p.p. is invariant under retractions. Hence every compact absolute retract hat f.*p.p. [S. Kinoshita, loc. cit.]. Further, if \(X\) and \(Y\) are two continua with f.*p.p. and \(X\cap Y\) is a single point, then \(X\cup Y\) has f.*p.p. [\textit{K. Borsuk}, Fundam. Math. 18, 198-213 (1932; Zbl 0004.22701); \textit{G. T. Whyburn}, Analytic topology, Am. Math. Soc. Colloq. Publ. 28 (1963; Zbl 0117.15804); the author and \textit{S. Kinoshita}, A property of essential components of the fixed point set, Kwansei Gakuin Univ. Ann. Stud. 37, 147-151 (1988)]. The latter statement has been extended to the special case where the number of continua is countably infinite [the author and Kinoshita, loc. cit.]. The purpose of this paper is to extend the above property to a more general setting; we prove that a continuum \(X\) has f.*p.p. whenever it can be expressed as the union of a null sequence of subcontinua \(X_\alpha\) with f.*p.p. such that any two of \(X_\alpha\) and \(X_\beta\) \((\alpha\neq\beta)\) have at most one point in common and the boundary of each component of \(X-X_\alpha\) consists of a single point for every \(\alpha\).