The oscillation function on metric spaces (Q1396796)

The aim of the paper is to prove the following assertion. Let \(X\) be a separable metric space which is \(c\)-dense (\(c=\text{card} \mathbb R)\) in itself. Let \(\{\Omega (y)\}_{y\in [0,1]}\) be a family of non-empty subsets of \(X\) such that: 1) The set \(\Omega (y)\) is closed for each \(y\in [0,1]\). 2) If \(y_1 < y_2\), then \(\Omega (y_2) \subset \Omega (y_1)\). 3) The set \(\bigcup _{y\in [0,1]} \Omega (y)\times \{y\}\) is closed in \(X\times \mathbb R\). 4) \(\Omega (0) = X\). Then there exists a function \(f\: X \rightarrow [0,1]\), such that \(\Omega (y) = \{x\in X\: \omega _f (x)\geq y\}\) for all \(y\in [0,1]\) (the symbol \(\omega _f(x)\) stands for the oscillation of the function \(f\) at the point \(x)\).