Editing Alexandroff extension (section)

== Further examples ==
=== Compactifications of discrete spaces===
* The one-point compactification of the set of positive integers is [[Homeomorphism|homeomorphic]] to the space consisting of ''K'' = {0} U {1/''n'' | ''n'' is a positive integer} with the order topology.
* A sequence <math>\{a_n\}</math> in a topological space <math>X</math> converges to a point <math>a</math> in <math>X</math>, if and only if the map <math>f\colon\mathbb N^*\to X</math> given by <math>f(n) = a_n</math> for <math>n</math> in <math>\mathbb N</math> and <math>f(\infty) = a</math> is continuous. Here <math>\mathbb N</math> has the [[discrete topology]].
* [[Polyadic space]]s are defined as topological spaces that are the continuous image of the power of a one-point compactification of a discrete, locally compact Hausdorff space.

=== Compactifications of continuous spaces===
* The one-point compactification of ''n''-dimensional Euclidean space '''R'''<sup>''n''</sup> is homeomorphic to the ''n''-sphere ''S''<sup>''n''</sup>.  As above, the map can be given explicitly as an ''n''-dimensional inverse stereographic projection.
* The one-point compactification of the product of <math>\kappa</math> copies of the half-closed interval [0,1), that is, of <math>[0,1)^\kappa</math>, is (homeomorphic to) <math>[0,1]^\kappa</math>. 
* Since the closure of a connected subset is connected, the Alexandroff extension of a noncompact connected space is connected.  However a one-point compactification may "connect" a disconnected space: for instance the one-point compactification of the disjoint union of a finite number <math>n</math> of copies of the interval (0,1) is a [[Bouquet of circles|wedge of <math>n</math> circles]].
* The one-point compactification of the disjoint union of a countable number of copies of the interval (0,1) is the [[Hawaiian earring]].  This is different from the wedge of countably many circles, which is not compact.
* Given <math>X</math> compact Hausdorff and <math>C</math> any closed subset of <math>X</math>, the one-point compactification of <math>X\setminus C</math> is <math>X/C</math>, where the forward slash denotes the [[quotient space (topology)|quotient space]].<ref name=rotman>[[Joseph J. Rotman]], ''An Introduction to Algebraic Topology'' (1988) Springer-Verlag {{ISBN|0-387-96678-1}} ''(See Chapter 11 for proof.)''</ref>
* If <math>X</math> and <math>Y</math> are locally compact Hausdorff, then <math>(X\times Y)^* = X^* \wedge Y^*</math> where <math>\wedge</math> is the [[smash product]].  Recall that the definition of the smash product:<math>A\wedge B = (A \times B) / (A \vee B)</math> where <math>A \vee B</math> is the [[wedge sum]], and again, / denotes the quotient space.<ref name=rotman/>

=== As a functor ===
The Alexandroff extension can be viewed as a [[functor]] from the [[category of topological spaces]] with proper continuous maps as morphisms to the category whose objects are continuous maps <math>c\colon X \rightarrow Y</math> and for which the morphisms from <math>c_1\colon X_1 \rightarrow Y_1</math> to <math>c_2\colon X_2 \rightarrow Y_2</math> are pairs of continuous maps <math>f_X\colon X_1 \rightarrow X_2, \ f_Y\colon  
Y_1 \rightarrow Y_2</math> such that <math>f_Y \circ c_1 = c_2 \circ f_X</math>.  In particular, homeomorphic spaces have isomorphic Alexandroff extensions.