Some pairs of manifolds with infinite uncountable \(\varphi\)-category (Q1423970)

The author studies the cardinality of the sets of critical points of smooth maps. A typical result is the following (Corollary 3.2): Let \(M,N\) be connected smooth manifolds such that \(\dim M \geq \dim N\). If \(f : M \to N\) is a non-surjective closed smooth mapping, then either all the points of \(M\) are critical or \(f\) has infinite uncountable number of critical values. In particular, if \(M\) is compact and \(N\) is non-compact then the critical point set has cardinality \(\aleph_1\).