On local compactness of the subspace FP2(X) of FP(X)

In this paper, it is shown that if X is homeomorphic to the topological sum of a Tychonoff space Y and a discrete space D  , then FP2(X)FP2(X) is locally compact if and only if FP2(Y)FP2(Y) is locally compact. Especially, the space FP2(X)FP2(X) is locally compact at its every point distinct from the identity e if X is homeomorphic to the topological sum of a compact space and a discrete space. Finally, a few questions about free paratopological groups are posed.