Alexandroff extension

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Short description: Way to extend a non-compact topological space

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

Motivation

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

The Alexandroff extension

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

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

The properties below follow from the above discussion:

  • The map c is continuous and open: it embeds X as an open subset of [math]\displaystyle{ X^* }[/math].
  • The space [math]\displaystyle{ X^* }[/math] is compact.
  • The image c(X) is dense in [math]\displaystyle{ X^* }[/math], if X is noncompact.
  • The space [math]\displaystyle{ X^* }[/math] is Hausdorff if and only if X is Hausdorff and locally compact.
  • The space [math]\displaystyle{ X^* }[/math] is T1 if and only if X is T1.

The one-point compactification

In particular, the Alexandroff extension [math]\displaystyle{ c: X \rightarrow X^* }[/math] 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 [math]\displaystyle{ X }[/math] is a compact Hausdorff space and [math]\displaystyle{ p }[/math] is a limit point of [math]\displaystyle{ X }[/math] (i.e. not an isolated point of [math]\displaystyle{ X }[/math]), [math]\displaystyle{ X }[/math] is the Alexandroff compactification of [math]\displaystyle{ X\setminus\{p\} }[/math].

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

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

  • The Alexandroff extension of [math]\displaystyle{ X }[/math] defined above. Here we take the complements of all closed compact subsets of [math]\displaystyle{ X }[/math] as neighborhoods of [math]\displaystyle{ \infty }[/math]. This is the largest topology that makes [math]\displaystyle{ X^* }[/math] a one-point compactification of [math]\displaystyle{ X }[/math].
  • The open extension topology. Here we add a single neighborhood of [math]\displaystyle{ \infty }[/math], namely the whole space [math]\displaystyle{ X^* }[/math]. This is the smallest topology that makes [math]\displaystyle{ X^* }[/math] a one-point compactification of [math]\displaystyle{ X }[/math].
  • Any topology intermediate between the two topologies above. For neighborhoods of [math]\displaystyle{ \infty }[/math] one has to pick a suitable subfamily of the complements of all closed compact subsets of [math]\displaystyle{ X }[/math]; 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 [math]\displaystyle{ \{a_n\} }[/math] in a topological space [math]\displaystyle{ X }[/math] converges to a point [math]\displaystyle{ a }[/math] in [math]\displaystyle{ X }[/math], if and only if the map [math]\displaystyle{ f\colon\mathbb N^*\to X }[/math] given by [math]\displaystyle{ f(n) = a_n }[/math] for [math]\displaystyle{ n }[/math] in [math]\displaystyle{ \mathbb N }[/math] and [math]\displaystyle{ f(\infty) = a }[/math] is continuous. Here [math]\displaystyle{ \mathbb N }[/math] 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 [math]\displaystyle{ \kappa }[/math] copies of the half-closed interval [0,1), that is, of [math]\displaystyle{ [0,1)^\kappa }[/math], is (homeomorphic to) [math]\displaystyle{ [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]\displaystyle{ n }[/math] of copies of the interval (0,1) is a wedge of [math]\displaystyle{ 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]\displaystyle{ X }[/math] compact Hausdorff and [math]\displaystyle{ C }[/math] any closed subset of [math]\displaystyle{ X }[/math], the one-point compactification of [math]\displaystyle{ X\setminus C }[/math] is [math]\displaystyle{ X/C }[/math], where the forward slash denotes the quotient space.[2]
  • If [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are locally compact Hausdorff, then [math]\displaystyle{ (X\times Y)^* = X^* \wedge Y^* }[/math] where [math]\displaystyle{ \wedge }[/math] is the smash product. Recall that the definition of the smash product:[math]\displaystyle{ A\wedge B = (A \times B) / (A \vee B) }[/math] where [math]\displaystyle{ A \vee B }[/math] 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 [math]\displaystyle{ c\colon X \rightarrow Y }[/math] and for which the morphisms from [math]\displaystyle{ c_1\colon X_1 \rightarrow Y_1 }[/math] to [math]\displaystyle{ c_2\colon X_2 \rightarrow Y_2 }[/math] are pairs of continuous maps [math]\displaystyle{ f_X\colon X_1 \rightarrow X_2, \ f_Y\colon Y_1 \rightarrow Y_2 }[/math] such that [math]\displaystyle{ f_Y \circ c_1 = c_2 \circ f_X }[/math]. In particular, homeomorphic spaces have isomorphic Alexandroff extensions.

See also

Notes

  1. ↑ "General topology – Non-Hausdorff one-point compactifications". https://math.stackexchange.com/questions/3817485/non-hausdorff-one-point-compactifications. 
  2. ↑ 2.0 2.1 Joseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag ISBN:0-387-96678-1 (See Chapter 11 for proof.)

References



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