Multi-regime feature of boiling system has ever eluded researchers despite decades of efforts. Basing on catastrophe theory, the topological boiling space was constructed from fundamental nucleating equation. All possible boiling curves are actually the projections of boiling space. It is demonstrated that transition of modes is virtually corresponded to the bifurcations. Catastrophe behaviors and hysteresis in boiling system have then been discussed by basic ideas in catastrophe theory. The effects of thermal physical properties and other parameters, such as liquid velocity, sub-cooled degree and gravity on boiling curves were revealed. The present model is in good agreement with available experimental facts.