In this paper, we prove the mean-convex neighborhood conjecture for neck singularities of mean curvature flow in $${\mathbb {R}}^{n+1}$$ all $$n\ge 3$$ : show that if a $$\{M_t\}$$ has an $$S^{n-1}\times {\mathbb {R}}$$ singularity at $$(x_0,t_0)$$ , then there exists $$\varepsilon =\varepsilon (x_0,t_0)>0$$ such $$M_t\cap B(x_0,\varepsilon )$$ is $$t\in (t_0-\varepsilon ^2,t_0+\varepsilon ^2)$...

The aim of this paper is to continue the study of (τ1, τ2) ∗-semi star generalized closed sets by introducing the concepts of (τ1, τ2) ∗-semi star generalized locally closed sets and study their basic properties in bitopological spaces. Keywords—(τ1, τ2) ∗-semi star generalized locally closed sets, τ1τ2-semi star generalized closed sets, (τ1, τ2) ∗-semi generalized locally closed sets, (τ1, τ2)...