Alexandroff extension

A geometrically appealing example of one-point compactification is given by the inverse stereographic projection. Recall that the stereographic projection S gives an explicit homeomorphism from the unit sphere minus the north pole (0,0,1) to the Euclidean plane. The inverse stereographic projection S^{-1}: \mathbb{R}^2 \hookrightarrow S^2 is an open, dense embedding into a compact Hausdorff space obtained by adjoining the additional point \infty = (0,0,1). Under the stereographic projection latitudinal circles z = c get mapped to planar circles r = \sqrt{(1+c)/(1-c)}. It follows that the deleted neighborhood basis of (0,0,1) given by the punctured spherical caps c \leq z corresponds to the complements of closed planar disks r \geq \sqrt{(1+c)/(1-c)}. More qualitatively, a neighborhood basis at \infty is furnished by the sets S^{-1}(\mathbb{R}^2 \setminus K) \cup \{ \infty \} as K ranges through the compact subsets of \mathbb{R}^2. This example already contains the key concepts of the general case. == Motivation ==

Let c: X \hookrightarrow Y be an embedding from a topological space X to a compact Hausdorff topological space Y, with dense image and one-point remainder \{ \infty \} = Y \setminus c(X). 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 \infty must be all sets obtained by adjoining \infty to the image under c of a subset of X with compact complement. == The Alexandroff extension ==

Let X be a topological space. Put X^* = X \cup \{\infty \}, and topologize X^* by taking as open sets all the open sets in X together with all sets of the form V = (X \setminus C) \cup \{\infty \} where C is closed and compact in X. Here, X \setminus C denotes the complement of C in X. Note that V is an open neighborhood of \infty, and thus any open cover of \{\infty \} will contain all except a compact subset C of X^*, implying that X^* is compact . The space X^* is called the Alexandroff extension of X (Willard, 19A). Sometimes the same name is used for the inclusion map c: X\to X^*. The properties below follow from the above discussion: • The map c is continuous and open: it embeds X as an open subset of X^*. • The space X^* is compact. • The image c(X) is dense in X^*, if X is noncompact. • The space X^* is Hausdorff if and only if X is Hausdorff and locally compact. • The space X^* is T1 if and only if X is T1. == The one-point compactification ==

In particular, the Alexandroff extension c: X \rightarrow X^* is a Hausdorff compactification of X if and only if X is Hausdorff, noncompact and locally compact. In this case it is called the one-point compactification or Alexandroff compactification of X. Recall from the above discussion that any Hausdorff compactification with one point remainder is necessarily (isomorphic to) the Alexandroff compactification. In particular, if X is a compact Hausdorff space and p is a limit point of X (i.e. not an isolated point of X), X is the Alexandroff compactification of X\setminus\{p\}. Let X be any noncompact Tychonoff space. Under the natural partial ordering on the set \mathcal{C}(X) of equivalence classes of compactifications, any minimal element is equivalent to the Alexandroff extension (Engelking, Theorem 3.5.12). It follows that a noncompact Tychonoff space admits a minimal compactification if and only if it is locally compact. == Non-Hausdorff one-point compactifications ==

Let (X,\tau) be an arbitrary noncompact topological space. One may want to determine all the compactifications (not necessarily Hausdorff) of X obtained by adding a single point, which could also be called one-point compactifications in this context. So one wants to determine all possible ways to give X^*=X\cup\{\infty\} a compact topology such that X is dense in it and the subspace topology on X induced from X^* is the same as the original topology. The last compatibility condition on the topology automatically implies that X is dense in X^*, because X is not compact, so it cannot be closed in a compact space. Also, it is a fact that the inclusion map c:X\to X^* is necessarily an open embedding, that is, X must be open in X^* and the topology on X^* must contain every member of \tau. So the topology on X^* is determined by the neighbourhoods of \infty. Any neighborhood of \infty is necessarily the complement in X^* of a closed compact subset of X, as previously discussed. The topologies on X^* that make it a compactification of X are as follows: • The Alexandroff extension of X defined above. Here we take the complements of all closed compact subsets of X as neighborhoods of \infty. This is the largest topology that makes X^* a one-point compactification of X. • The open extension topology. Here we add a single neighborhood of \infty, namely the whole space X^*. This is the smallest topology that makes X^* a one-point compactification of X. • Any topology intermediate between the two topologies above. For neighborhoods of \infty one has to pick a suitable subfamily of the complements of all closed compact subsets of X; for example, the complements of all finite closed compact subsets, or the complements of all countable closed compact subsets. == Further examples ==

Compactifications of discrete spaces • The one-point compactification of the set of positive integers is homeomorphic to the space consisting of K = {0} U {1/n | n is a positive integer} with the order topology. • A sequence \{a_n\} in a topological space X converges to a point a in X, if and only if the map f\colon\mathbb N^*\to X given by f(n) = a_n for n in \mathbb N and f(\infty) = a is continuous. Here \mathbb N has the discrete topology. • Polyadic spaces 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 Rn is homeomorphic to the n-sphere Sn. As above, the map can be given explicitly as an n-dimensional inverse stereographic projection. • The one-point compactification of the product of \kappa copies of the half-closed interval [0,1), that is, of [0,1)^\kappa, is (homeomorphic to) [0,1]^\kappa. • 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 n of copies of the interval (0,1) is a wedge of n 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 X compact Hausdorff and C any closed subset of X, the one-point compactification of X\setminus C is X/C, where the forward slash denotes the quotient space. • If X and Y are locally compact Hausdorff, then (X\times Y)^* = X^* \wedge Y^* where \wedge is the smash product. Recall that the definition of the smash product:A\wedge B = (A \times B) / (A \vee B) where A \vee B is the wedge sum, and again, / denotes the quotient space. 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 c\colon X \rightarrow Y and for which the morphisms from c_1\colon X_1 \rightarrow Y_1 to c_2\colon X_2 \rightarrow Y_2 are pairs of continuous maps f_X\colon X_1 \rightarrow X_2, \ f_Y\colon Y_1 \rightarrow Y_2 such that f_Y \circ c_1 = c_2 \circ f_X. In particular, homeomorphic spaces have isomorphic Alexandroff extensions. The latter is the arrow category of topological spaces, often constructed as category of functors from interval category \operatorname{Arr}(\mathrm{Top}) = \operatorname{Func}(\mathrm{I}, \mathrm{Top}), where interval category is the category with 2 objects connected by single arrow. == See also ==