Alexandroff extension explained

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 ^{-1} : R ^{2} \hookrightarrow S ^{2} is an open, dense embedding into a compact Hausdorff space obtained by adjoining the additional point inf ty = ( 0 , 0 , 1 ) . Under the stereographic projection latitudinal circles z = c get mapped to planar circles $r\; =\; \backslash sqrt$. 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 $r\; \backslash geq\; \backslash sqrt$. More qualitatively, a neighborhood basis at inf ty is furnished by the sets S ^{-1} ( R ^{2} \setminus K) \cup \{ inf ty \} as K ranges through the compact subsets of 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 \{ inf ty \} = 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 inf ty must be all sets obtained by adjoining inf ty 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 \{ inf ty \}, 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 \{ inf ty \} 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 inf ty, and thus any open cover of \{ inf ty \} 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 → 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

l{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 memberof \tau .^[1] So the topology on X ^{*} is determined by the neighbourhoods of inf ty . Any neighborhood of inf ty 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 inf ty . 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

inf ty , 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

inf ty 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

\{a _{n\}} in a topological space X converges to a point a in X , if and only if the map f\colon N ^{*\to} X given by f(n) = a _{n} for n in N and f( inf ty) = a is continuous. Here 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 <math>[0,1)^\kappa</math>, is (homeomorphic to) <math>[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.^[2]

• If

X and Y are locally compact Hausdorff, then (X x Y) ^{*} = X ^{*} \wedge Y ^{*} where \wedge is the smash product. Recall that the definition of the smash product:A\wedge B = (A x B) / (A \vee B) where A \vee B is the wedge sum, and again, / denotes the quotient space.^[2]

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 → Y and for which the morphisms from c _{1\colon} X _{1} → Y _{1} to c _{2\colon} X _{2} → Y _{2} are pairs of continuous maps f _{X\colon} X _{1} → X _{2,} f _{Y\colon} Y _{1} → Y _{2} such that f _{Y} \circ c _{1} = c _{2} \circ f _{X} . In particular, homeomorphic spaces have isomorphic Alexandroff extensions.

Notes and References

1. Web site: General topology – Non-Hausdorff one-point compactifications.
2. [Joseph J. Rotman]