Locally connected curves admit small retractions onto graphs (Q2901927)

For an arbitrary \(\varepsilon >0\), an \(\varepsilon\)-mapping of a metric space \((X,d)\) onto a set \(Y\subset X\) is a continuous function \(f:X\rightarrow Y\) such that diameter\((f^{-1}(y))<\varepsilon \) for all \(y\in Y\). A curve is a one-dimensional metric continuum. An old result by \textit{S. Mazurkiewicz} [Fundam. Math. 20, 281--284 (1933; Zbl 0006.42601)] says that for every locally connected curve \(X\) and every \(\varepsilon >0\) there is an \(\varepsilon\)-mapping \(f:X\rightarrow X\) such that \(f(X)\) is a finite graph. In the paper under review, the author improves this result by proving that the mapping can be also asked to be a retraction.