When the family of functions vanishing at infinity is an ideal of \(C(X)\). (Q1414912)

The authors characterize Tychonoff spaces \(X\) such that the set \(C_\infty(X)=\{f\in C(X):\{x\in X:| f(x) | \geq\varepsilon\}\) is compact for each \(\varepsilon>0\}\) is an ideal in the ring \(C(X)\) of all continuous real-valued functions on \(X\). This happens if and only if each open locally compact subset \(B\) of \(X\) is bounded (in the sense that \(f(B)\) is bounded for any \(f\in C(X)\)). Another result of the paper asserts that \(C_\infty(X)=C_K(X) \), the ideal of functions with compact support, iff each open locally compact \(\sigma\)-compact subset of \(X\) has compact closure.