Let be a topological space. A topological space is called an extension of if contain as a dense suace. An extension Yof is called a one-point extension if is a single point. One-point extension has been first introduced by P. Alexandroff in 1924 when he showed that every locally compact non-compact Hausdorff space has a one-point compact extension called the onepoint compactification of or the Alexandroff compactification of The Alexandroff construction of the one-point compactification has been further generalized by various authors in various directions. In particular, there has been lots of interest in the problem of whether a space having a topological property locally has a one-point extension having the topological property globally. Here we discuss such generalizations through the introduction of the notion of boundedness. More precisely we have the following. Let be a topological space. A family of subset of is called a boundedness in if it .