Phase-transitions are normally known from physics, e.g. when a matter changes its state of aggregation while an appropriate transition-threshold changes into a decisive quality. Phenomena of such kinds are not reserved for physics only, they can also be observed - detached from any physical application - in a wide range of mathematical contexts. These scenarios - generally called as phase-transitions now - will mediate between manifolds of different topologies. Some physics-examples are known where the threshold of the appropriate transition is excelled by a fractal structure with the property of invariance in renormalizations. Distribution of magnetized micro-cells in a Ferro-magnet at a critical temperature may be mentioned as a typical example in this sense. Similar qualities can also be verified for phase-transitions in pure mathematical contexts, especially when the transitions between manifolds of different topologies are mediated by fractals that turn out to be invariant in renormalizations. Therefore it seems, this kind of phenomena will always happen in space as soon the aforementioned conditions are met.