Local topology of foliation spaces on surfaces of closed varieties of dimension 3 (Q878324)

Let \(M\) be a closed orientable 3-manifold and \(\mathbb{F}_1(M)\) the space of orientable foliations of codimension 1 of \(M\) with \(C^\infty\)-topology. The author proves the following theorem: Let \(\mathcal{F}\in\mathbb{F}_1(M)\) be a topologically taut foliation. Then there exists a neighbourhood \(V(\mathcal{F})\) of \(\mathcal{F}\) such that for each foliation \(\mathcal{F}'\in V(\mathcal{F})\) there exists a continuous mapping \(f:[0,1]\to\mathbb{F}_1(M)\) fulfilling the conditions: \(f(0) =\mathcal{F}\) and \(f(1) =\mathcal{F}'\).