  Let $M$ be a smooth closed orientable surface, and let $F$ be the space of
Morse functions on $M$ such that at least $\chi(M)+1$ critical points of each
function of $F$ are labeled by different labels (enumerated). Endow the space
$F$ with $C^\infty$-topology. We prove the homotopy equivalence $F\sim
R\times{\widetilde{\cal M}}$ where $R$ is one of the manifolds ${\mathbb
R}P^3$, $S^1\times S^1$ and the point in dependence on the sign of $\chi(M)$,
and ${\widetilde{\cal M}}$ is the universal moduli space of framed Morse
functions, which is a smooth stratified manifold. Morse inequalities for the
Betti numbers of the space $F$ are obtained.

Geometric Topology;Algebraic Topology