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Alexandroff extension

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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).

Contents

• Example: inverse stereographic projection

• Motivation

• The Alexandroff extension

• The one-point compactification

• Non-Hausdorff one-point compactifications

• Further examples

• Compactifications of discrete spaces

• Compactifications of continuous spaces

• As a functor

• See also

• Notes

• References

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 ↪ ↪ S 2 {\displaystyle S^{-1}:\mathbb {R} ^{2}\hookrightarrow S^{2}} is an open, dense embedding into a compact Hausdorff space obtained by adjoining the additional point ∞ ∞ = ( 0 , 0 , 1 ) {\displaystyle \infty =(0,0,1)} . Under the stereographic projection latitudinal circles z = c {\displaystyle z=c} get mapped to planar circles r = ( 1 + c ) / ( 1 − − c ) {\textstyle r={\sqrt {(1+c)/(1-c)}}} . It follows that the deleted neighborhood basis of ( 0 , 0 , 1 ) {\displaystyle (0,0,1)} given by the punctured spherical caps c ≤ ≤ z < 1 {\displaystyle c\leq z<1} corresponds to the complements of closed planar disks r ≥ ≥ ( 1 + c ) / ( 1 − − c ) {\textstyle r\geq {\sqrt {(1+c)/(1-c)}}} . More qualitatively, a neighborhood basis at ∞ ∞ {\displaystyle \infty } is furnished by the sets S − − 1 ( R 2 ∖ ∖ K ) ∪ ∪ { ∞ ∞ } {\displaystyle S^{-1}(\mathbb {R} ^{2}\setminus K)\cup \{\infty \}} as K ranges through the compact subsets of R 2 {\displaystyle \mathbb {R} ^{2}} . This example already contains the key concepts of the general case.

Motivation

Let c : X ↪ ↪ Y {\displaystyle 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 ∖ ∖ c ( X ) {\displaystyle \{\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 ∞ ∞ {\displaystyle \infty } must be all sets obtained by adjoining ∞ ∞ {\displaystyle \infty } to the image under c of a subset of X with compact complement.

The Alexandroff extension

Let X {\displaystyle X} be a topological space. Put X ∗ ∗ = X ∪ ∪ { ∞ ∞ } , {\displaystyle X^{*}=X\cup \{\infty \},} and topologize X ∗ ∗ {\displaystyle X^{*}} by taking as open sets all the open sets in X together with all sets of the form V = ( X ∖ ∖ C ) ∪ ∪ { ∞ ∞ } {\displaystyle V=(X\setminus C)\cup \{\infty \}} where C is closed and compact in X. Here, X ∖ ∖ C {\displaystyle X\setminus C} denotes the complement of C {\displaystyle C} in X . {\displaystyle X.} Note that V {\displaystyle V} is an open neighborhood of ∞ ∞ , {\displaystyle \infty ,} and thus any open cover of { ∞ ∞ } {\displaystyle \{\infty \}} will contain all except a compact subset C {\displaystyle C} of X ∗ ∗ , {\displaystyle X^{*},} implying that X ∗ ∗ {\displaystyle X^{*}} is compact (Kelley 1975, p. 150).

The space X ∗ ∗ {\displaystyle X^{*}} is called the Alexandroff extension of X (Willard, 19A). Sometimes the same name is used for the inclusion map c : X → → X ∗ ∗ . {\displaystyle 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 ∗ ∗ {\displaystyle X^{*}} .

• The space X ∗ ∗ {\displaystyle X^{*}} is compact.

• The image c(X) is dense in X ∗ ∗ {\displaystyle X^{*}} , if X is noncompact.

• The space X ∗ ∗ {\displaystyle X^{*}} is Hausdorff if and only if X is Hausdorff and locally compact.

• The space X ∗ ∗ {\displaystyle X^{*}} is T1 if and only if X is T1.

The one-point compactification

In particular, the Alexandroff extension c : X → → X ∗ ∗ {\displaystyle 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 {\displaystyle X} is a compact Hausdorff space and p {\displaystyle p} is a limit point of X {\displaystyle X} (i.e. not an isolated point of X {\displaystyle X} ), X {\displaystyle X} is the Alexandroff compactification of X ∖ ∖ { p } {\displaystyle X\setminus \{p\}} .

Let X be any noncompact Tychonoff space. Under the natural partial ordering on the set C ( X ) {\displaystyle {\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 , τ τ ) {\displaystyle (X,\tau )} be an arbitrary noncompact topological space. One may want to determine all the compactifications (not necessarily Hausdorff) of X {\displaystyle 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 ∪ ∪ { ∞ ∞ } {\displaystyle X^{*}=X\cup \{\infty \}} a compact topology such that X {\displaystyle X} is dense in it and the subspace topology on X {\displaystyle X} induced from X ∗ ∗ {\displaystyle X^{*}} is the same as the original topology. The last compatibility condition on the topology automatically implies that X {\displaystyle X} is dense in X ∗ ∗ {\displaystyle X^{*}} , because X {\displaystyle X} is not compact, so it cannot be closed in a compact space. Also, it is a fact that the inclusion map c : X → → X ∗ ∗ {\displaystyle c:X\to X^{*}} is necessarily an open embedding, that is, X {\displaystyle X} must be open in X ∗ ∗ {\displaystyle X^{*}} and the topology on X ∗ ∗ {\displaystyle X^{*}} must contain every member of τ τ {\displaystyle \tau } .cite-ref-1[1] So the topology on X ∗ ∗ {\displaystyle X^{*}} is determined by the neighbourhoods of ∞ ∞ {\displaystyle \infty } . Any neighborhood of ∞ ∞ {\displaystyle \infty } is necessarily the complement in X ∗ ∗ {\displaystyle X^{*}} of a closed compact subset of X {\displaystyle X} , as previously discussed.

The topologies on X ∗ ∗ {\displaystyle X^{*}} that make it a compactification of X {\displaystyle X} are as follows:

• The Alexandroff extension of X {\displaystyle X} defined above. Here we take the complements of all closed compact subsets of X {\displaystyle X} as neighborhoods of ∞ ∞ {\displaystyle \infty } . This is the largest topology that makes X ∗ ∗ {\displaystyle X^{*}} a one-point compactification of X {\displaystyle X} .

• The open extension topology. Here we add a single neighborhood of ∞ ∞ {\displaystyle \infty } , namely the whole space X ∗ ∗ {\displaystyle X^{*}} . This is the smallest topology that makes X ∗ ∗ {\displaystyle X^{*}} a one-point compactification of X {\displaystyle X} .

• Any topology intermediate between the two topologies above. For neighborhoods of ∞ ∞ {\displaystyle \infty } one has to pick a suitable subfamily of the complements of all closed compact subsets of X {\displaystyle 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 } {\displaystyle \{a_{n}\}} in a topological space X {\displaystyle X} converges to a point a {\displaystyle a} in X {\displaystyle X} , if and only if the map f : : N ∗ ∗ → → X {\displaystyle f\colon \mathbb {N} ^{*}\to X} given by f ( n ) = a n {\displaystyle f(n)=a_{n}} for n {\displaystyle n} in N {\displaystyle \mathbb {N} } and f ( ∞ ∞ ) = a {\displaystyle f(\infty )=a} is continuous. Here N {\displaystyle \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 κ κ {\displaystyle \kappa } copies of the half-closed interval [0,1), that is, of [ 0 , 1 ) κ κ {\displaystyle [0,1)^{\kappa }} , is (homeomorphic to) [ 0 , 1 ] κ κ {\displaystyle [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 {\displaystyle n} of copies of the interval (0,1) is a wedge of n {\displaystyle 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 {\displaystyle X} compact Hausdorff and C {\displaystyle C} any closed subset of X {\displaystyle X} , the one-point compactification of X ∖ ∖ C {\displaystyle X\setminus C} is X / C {\displaystyle X/C} , where the forward slash denotes the quotient space.cite-ref-rotman-2-0[2]

• If X {\displaystyle X} and Y {\displaystyle Y} are locally compact Hausdorff, then ( X × × Y ) ∗ ∗ = X ∗ ∗ ∧ ∧ Y ∗ ∗ {\displaystyle (X\times Y)^{*}=X^{*}\wedge Y^{*}} where ∧ ∧ {\displaystyle \wedge } is the smash product. Recall that the definition of the smash product: A ∧ ∧ B = ( A × × B ) / ( A ∨ ∨ B ) {\displaystyle A\wedge B=(A\times B)/(A\vee B)} where A ∨ ∨ B {\displaystyle A\vee B} is the wedge sum, and again, / denotes the quotient space.cite-ref-rotman-2-1[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 : : X → → Y {\displaystyle c\colon X\rightarrow Y} and for which the morphisms from c 1 : : X 1 → → Y 1 {\displaystyle c_{1}\colon X_{1}\rightarrow Y_{1}} to c 2 : : X 2 → → Y 2 {\displaystyle c_{2}\colon X_{2}\rightarrow Y_{2}} are pairs of continuous maps f X : : X 1 → → X 2 , f Y : : Y 1 → → Y 2 {\displaystyle f_{X}\colon X_{1}\rightarrow X_{2},\ f_{Y}\colon Y_{1}\rightarrow Y_{2}} such that f Y ∘ ∘ c 1 = c 2 ∘ ∘ f X {\displaystyle f_{Y}\circ c_{1}=c_{2}\circ f_{X}} . In particular, homeomorphic spaces have isomorphic Alexandroff extensions.