Editing Alexandroff extension (section)

== Motivation ==
Let <math>c: X \hookrightarrow Y</math> be an embedding from a topological space ''X'' to a compact Hausdorff topological space ''Y'', with dense image and one-point remainder <math>\{ \infty \} = Y \setminus c(X)</math>.  Then ''c''(''X'') is open in a compact Hausdorff space so is locally compact Hausdorff, hence its homeomorphic preimage ''X'' is also locally compact Hausdorff.  Moreover, if ''X'' were compact then ''c''(''X'') would be closed in ''Y'' and hence not dense. Thus a space can only admit a Hausdorff one-point compactification if it is locally compact, noncompact and Hausdorff.  Moreover, in such a one-point compactification the image of a neighborhood basis for ''x'' in ''X'' gives a neighborhood basis for ''c''(''x'') in ''c''(''X''), and—because a subset of a compact Hausdorff space is compact if and only if it is closed—the open neighborhoods of <math>\infty</math> must be all sets obtained by adjoining <math>\infty</math> to the image under ''c'' of a subset of ''X'' with compact complement.