Bel’nov has studied the partially ordered set long time ago. The authors have been unable to locate any other research about in the literature. There is a well-known result of Magill that relates the partially ordered set of all compactification of a completelly regular space to the topology of the outgrowth (where is the Stone-?ech compactification of ). Motivated by Magill’s result and Bel’nov study of Henriksen, Janos and Woods studied the partially ordered set of . Although their study seems to be more related to that of Bel’nov, the nature of result resemble more to those obtained by Magill. They have defined an order anti-isomorphism from the set of all one-point metrizable extension of a locally compact metrizable space to the set of all zeroset of (partially ordered by set inclusion). The mapping enables us to study y studying its image . In particular, the authors proved that an element of is locally compact if and only if its image under is an open-closed (clopen) subset of <v:imagedata src="file:///D:\\DOCUME~1\\mirzai\\LOCALS~1\emp\\msohtmlclip1\\01\\clip_image027.png" chromakey="white" o:title="