We study the spherically symmetric collapse of a fluid with non-vanishing radial pressure in higher dimensional space-time. We obtain the general exact solution in the closed form for the equation of state ($P_r = \gamma \rho$) which leads to the explicit construction of the root equation governing the nature (black hole versus naked singularity) of the central singularity. A remarkable feature of the root equation is its invariance for the three cases: (${ D+1}, { \gamma = -1}$), (${ D}, { \gamma = 0}$) and (${ D - 1}, { \gamma = 1}$) where $D$ is the dimension of space-time. That is, for the ultimate end result of the collapse, $D$-dimensional dust, ${ D+1}$ - AdS (anti de Sitter)-like and ${ D-1}$ - dS-like are absolutely equivalent.
Categories: gr-qc