Alexandroff one-point compactification

In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician Pavel Alexandroff.
More precisely, let ''X'' be a topological space. Then the Alexandroff extension of ''X'' is a certain compact space ''X''* together with an open embedding ''c'' : ''X'' → ''X''* such that the complement of ''X'' in ''X''* consists of a single point, typically denoted ∞. The map ''c'' is a Hausdorff compactification if and only if ''X'' is a locally compact, noncompact Hausdorff space. For such spaces the Alexandroff extension is called the one-point compactification or Alexandroff compactification. The advantages of the Alexandroff compactification lie in its simple, often geometrically meaningful structure and the fact that it is in a precise sense minimal among all compactifications; the disadvantage lies in the fact that it only gives a Hausdorff compactification on the class of locally compact, noncompact Hausdorff spaces, unlike the Stone–Čech compactification which exists for any topological space (but provides an embedding exactly for Tychonoff spaces).

Example: inverse stereographic projection

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 $S^: \mathbb^2 \hookrightarrow S^2$ 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 $r = \sqrt$. It follows that the deleted neighborhood basis of $(0,0,1)$ given by the punctured spherical caps $c \leq z < 1$ corresponds to the complements of closed planar disks $r \geq \sqrt$. More qualitatively, a neighborhood basis at $\infty$ is furnished by the sets $S^(\mathbb^2 \setminus K) \cup \$ as ''K'' ranges through the compact subsets of $\mathbb^2$. 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 $\ = Y \setminus c(X)$. 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 $X^* = X \cup \,$ 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 \$ 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 $\$ 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\$.

Let ''X'' be any noncompact Tychonoff space. Under the natural partial ordering on the set $\mathcal(X)$ 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\$ 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'' = U with the order topology.
* A sequence $\$ 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 R^''n'' is homeomorphic to the ''n''-sphere ''S''^''n''. 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 ,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 Bouquet of circles">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.Joseph J. Rotman">quotient space (topology)">quotient space.Joseph J. Rotman, ''An Introduction to Algebraic Topology'' (1988) Springer-Verlag ''(See Chapter 11 for proof.)''
* 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.

See also

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Notes

References

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* {{Citation , last=Willard , first=Stephen , title=General Topology , publisher=Addison-Wesley , isbn=3-88538-006-4 , mr=0264581 , zbl=0205.26601 , year=1970

General topology
Compactification (mathematics)