Alexandroff extension

Template:Short description 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 projectionEdit

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>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>\infty = (0,0,1)</math>. Under the stereographic projection latitudinal circles <math>z = c</math> get mapped to planar circles <math display=inline>r = \sqrt{(1+c)/(1-c)}</math>. It follows that the deleted neighborhood basis of <math>(0,0,1)</math> given by the punctured spherical caps <math>c \leq z < 1</math> corresponds to the complements of closed planar disks <math display=inline>r \geq \sqrt{(1+c)/(1-c)}</math>. More qualitatively, a neighborhood basis at <math>\infty</math> is furnished by the sets <math>S^{-1}(\mathbb{R}^2 \setminus K) \cup \{ \infty \}</math> as K ranges through the compact subsets of <math>\mathbb{R}^2</math>. This example already contains the key concepts of the general case.

MotivationEdit

Let <math>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>\{ \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>\infty</math> must be all sets obtained by adjoining <math>\infty</math> to the image under c of a subset of X with compact complement.

The Alexandroff extensionEdit

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

The space <math>X^*</math> is called the Alexandroff extension of X (Willard, 19A). Sometimes the same name is used for the inclusion map <math>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>X^*</math>.
The space <math>X^*</math> is compact.
The image c(X) is dense in <math>X^*</math>, if X is noncompact.
The space <math>X^*</math> is Hausdorff if and only if X is Hausdorff and locally compact.
The space <math>X^*</math> is T₁ if and only if X is T₁.

The one-point compactificationEdit

In particular, the Alexandroff extension <math>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>X</math> is a compact Hausdorff space and <math>p</math> is a limit point of <math>X</math> (i.e. not an isolated point of <math>X</math>), <math>X</math> is the Alexandroff compactification of <math>X\setminus\{p\}</math>.

Let X be any noncompact Tychonoff space. Under the natural partial ordering on the set <math>\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 compactificationsEdit

Let <math>(X,\tau)</math> be an arbitrary noncompact topological space. One may want to determine all the compactifications (not necessarily Hausdorff) of <math>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>X^*=X\cup\{\infty\}</math> a compact topology such that <math>X</math> is dense in it and the subspace topology on <math>X</math> induced from <math>X^*</math> is the same as the original topology. The last compatibility condition on the topology automatically implies that <math>X</math> is dense in <math>X^*</math>, because <math>X</math> is not compact, so it cannot be closed in a compact space. Also, it is a fact that the inclusion map <math>c:X\to X^*</math> is necessarily an open embedding, that is, <math>X</math> must be open in <math>X^*</math> and the topology on <math>X^*</math> must contain every member of <math>\tau</math>.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> So the topology on <math>X^*</math> is determined by the neighbourhoods of <math>\infty</math>. Any neighborhood of <math>\infty</math> is necessarily the complement in <math>X^*</math> of a closed compact subset of <math>X</math>, as previously discussed.

The topologies on <math>X^*</math> that make it a compactification of <math>X</math> are as follows:

The Alexandroff extension of <math>X</math> defined above. Here we take the complements of all closed compact subsets of <math>X</math> as neighborhoods of <math>\infty</math>. This is the largest topology that makes <math>X^*</math> a one-point compactification of <math>X</math>.
The open extension topology. Here we add a single neighborhood of <math>\infty</math>, namely the whole space <math>X^*</math>. This is the smallest topology that makes <math>X^*</math> a one-point compactification of <math>X</math>.
Any topology intermediate between the two topologies above. For neighborhoods of <math>\infty</math> one has to pick a suitable subfamily of the complements of all closed compact subsets of <math>X</math>; for example, the complements of all finite closed compact subsets, or the complements of all countable closed compact subsets.

Further examplesEdit

Compactifications of discrete spacesEdit

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>\{a_n\}</math> in a topological space <math>X</math> converges to a point <math>a</math> in <math>X</math>, if and only if the map <math>f\colon\mathbb N^*\to X</math> given by <math>f(n) = a_n</math> for <math>n</math> in <math>\mathbb N</math> and <math>f(\infty) = a</math> is continuous. Here <math>\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 spacesEdit

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>\kappa</math> copies of the half-closed interval [0,1), that is, of <math>[0,1)^\kappa</math>, is (homeomorphic to) <math>[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>n</math> of copies of the interval (0,1) is a wedge of <math>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>X</math> compact Hausdorff and <math>C</math> any closed subset of <math>X</math>, the one-point compactification of <math>X\setminus C</math> is <math>X/C</math>, where the forward slash denotes the quotient space.<ref name=rotman>Joseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag Template:ISBN (See Chapter 11 for proof.)</ref>
If <math>X</math> and <math>Y</math> are locally compact Hausdorff, then <math>(X\times Y)^* = X^* \wedge Y^*</math> where <math>\wedge</math> is the smash product. Recall that the definition of the smash product:<math>A\wedge B = (A \times B) / (A \vee B)</math> where <math>A \vee B</math> is the wedge sum, and again, / denotes the quotient space.<ref name=rotman/>

As a functorEdit

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>c\colon X \rightarrow Y</math> and for which the morphisms from <math>c_1\colon X_1 \rightarrow Y_1</math> to <math>c_2\colon X_2 \rightarrow Y_2</math> are pairs of continuous maps <math>f_X\colon X_1 \rightarrow X_2, \ f_Y\colon Y_1 \rightarrow Y_2</math> such that <math>f_Y \circ c_1 = c_2 \circ f_X</math>. In particular, homeomorphic spaces have isomorphic Alexandroff extensions.

NotesEdit

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