One-point extensions of locally compact paracompact spaces

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 {em 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 leq Y , if there is a continuous function of Y into Y 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 beta X / X. Here P is subject to some requirements and include sigma-compactness and the Lindelöf property as special cases.