Every continuous real-valued function defined on a connected and locally connected topological space is a quotient function (Q1120143)

A mapping f: \(X\to Y\) is said to be compact-covering if for every compact set B of f(X) there is a compact subset A of X such that \(f(A)=B\). \textit{A. Villani} [Boll. Unione Mat. Ital., VI. Ser., B 3, 99-109 (1984; Zbl 0541.54023)] has proved that each continuous real-valued function f: \(M\to R\) defined on a connected, locally connected complete metric space M is compact-covering, and he has asked if the result is true without the assumption of completeness of M. \textit{A. V. Arkhangel'skij} [Dokl. Akad. Nauk SSSR 155, 247-250 (1964; Zbl 0129.381)] has shown that each compact-covering mapping is quotient, and has asked in his review of Villani's paper in RŽ Mat. (1985) A 538, if, again without completeness of M, it can be shown that f is quotient. The author gives an affirmative answer to this question by showing the statement formulated in the title. Furthermore, he presents an example showing a negative answer to the question asked by Villani. \{Reviewer's remark: A wider treatment of the topic of the reviewed paper can be found in K. Omiljanowski, Boll. Unione Mat. Ital. (7) 1-B, 649-661 (1987)]. The reader is also referred to a large paper by the reviewer, the author, \textit{B. Ricceri} and \textit{A. Villani} in Rend. Circ. Mat. Palermo, II. Ser. 37, No.2, 261-281 (1988).\}