The aim of this paper is to introduce and give preliminary investigation of ∂-partial-Locally compact spaces. Locally compactness and ∂-partial-Locally compactness are independent of each other. Every Locally compact Hausdorff space is ∂-partial-Locally compact. But the converse is not true even though it be Hausdorff. ∂-partial-Locally compactness is a topological property. ∂-partial-Locally compactness is not preserved by the product topology.

We define a new function-valued inner product on L2(G), called j-bracket product, where G is a Locally compact abelian group and j is a topological isomorphism on G. We investigate the notion of j -orthogonality, Bessel's Inequality and j -orthonormal bases with respect to this inner product on L2(G).

A space Y is called an extension of a space X, if Y contains X as a dense subspace. Two extensions of X are said to be equivalent, if there is a homeomorphism between them which fixes X point-wise. For two (equivalence classes of) extensions Y and Y’ of X let Y£ Y’, if there is a continuous function of Y 0 intoY which fixes X point-wise. An extension Y of X is called a one-point extension, if Y \X is a singleton. An extension Y of X is called first-countable, if Y is first-countable at points of Y \X. Let P be a topological property. An extension Y of X is called a P-extension, if it has P.In this article, for a given Locally compact paracompact space X, we consider the two classes of one-point Cech-complete, P extensions of X and one-point first-countable Locally-P extensions of X, and we study their order-structures, by relating them to the topology of a certain subspace of the outgrowth bX\X. Here P is subject to some requirements and include s-compactness and the Lindelof property as special cases.

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