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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Let K be a Locally compact hypergroup. In this paper we initiate the concept of fundamental domain in Locally compact hypergroups and then we introduce the Borel section mapping. In fact, a fundamental domain is a subset of a hypergroup K including a unique element from each cosets, and the Borel section mapping is a function which corresponds to any coset, the related unique element in the fundamental domain. Finally, as an application we show that if K is a Locally compact hypergroup and H is one of its commutative subhypergroup, then there exists an isometric transform Z from L^2 (K) to L^2 (H ̂ , L^2 (H\K)). For this, we apply the dual of hypergroups and specially we use the Plancherel Theorem. This transform is a version of the Zak transform on Locally compact hypergroups which can be considered as an extension of the usual notion of Zak transform in the case of Locally compact groups.