The fuzzy topological space was introduced by Dip in 1999 depending on the notion of the fuzzy space (X, I). In this paper, further definitions, and theorems on The fuzzy topological space was introduced by Dip in 1999 depending on the notion of the fuzzy space (X, I). In this paper, further definitions, and theorems on

We find the image of the affine symplectic Lie algebra gn from the Leibniz homology HL⁎(gn) to the Lie algebra homology H⁎Lie(gn). The result shows that the image is the exterior algebra ∧⁎(wn) generated by the forms wn=∑i=1n(∂/∂xi∧∂/∂yi). Given the relevance of Hochschild homology to string topology and to get more interesting applications, we show that such a map is of potential interest in string topology and homological algebra by taking into account that the Hochschild homology HH⁎-1(U(gn)) is isomorphic to H⁎-1Lie(gn,U(gn)ad). Explicitly, we use the alternation of multilinear map, in our elements, to do certain calculations.

We construct and study the map from Leibniz homology HL_∗(