strongly pointwise self-homeomorphic

A topological space X is called strongly pointwise self-homeomorphic at a point $x \in X$ if for any neighborhood U of x there is a neighborhood V of x such that $x\in V \subseteq U$ and V is homeomorphic to X. The space X is called strongly pointwise self-homeomorphic if it is strongly pointwise self-homeomorphic at each of its points.