Hausdorff Compactifications
aa r X i v : . [ m a t h . GN ] S e p HAUSDORFF COMPACTIFICATIONS
MATT INSALL, PETER A. LOEB, AND MA LGORZATA ANETA MARCINIAK
Abstract.
Previously, the authors used the insights of Robinson’s non-standard analysis as a powerful tool to extend and simplify the construc-tion of some compactifications of regular spaces. They now show thatany Hausdorff compactification is obtainable with their method. Introduction
In [3], we extended to more general compactifications the work on topo-logical ends of Insall and Marciniak in [2]. Fix an appropriately saturatednonstandard extension of a regular, noncompact topological space ( X, T ). In[3], we showed that points needed to form a compactification can be formedfrom equivalence classes of points not in the monad of any standard point.Any equivalence relation on such points works, but not every compactifi-cation of X can be obtained this way. We show here that any Hausdorffcompactification of X can be obtained using a natural equivalence relation.We conclude with brief discussions of an application to a moduli space oftriangles and to the Martin boundary in potential theory and probability.2. General Compactifications
First we review the construction in [3]. Also for background, see [8], [9],and [7]. Let ( X, T ) be a regular, noncompact topological space.2.1. Definition.
By a compactification of ( X, T ) , we mean a compact space ( Y, T Y ) such that X is a dense subset of ( Y, T Y ) . We require here that theidentity mapping of X into Y is a homeomorphism from ( X, T ) to X withthe relative T Y -topology. Example.
A simple example where the latter requirement fails is givenby the set X = { /n : n ∈ N } ∪ { } with the discrete topology. We set Y = X , but now the singleton { } is no longer an open set. Its neighborhoodsconsist of all intervals in X from 0 to 1 /n , n ∈ N . Clearly, the set X is adense and continuous image in Y , but Y is not a compactification of X inthe sense of Definition 2.1. Date : September 4, 2020.2010
Mathematics Subject Classification.
Primary 03H05, 54D35; Secondary 54J05.
Key words and phrases.
Compactifications, Hausdorff compactifications, nonstandardmethods, moduli space.
We now fix a κ -saturated nonstandard extension of ( X, T ), where κ isgreater than the cardinality of the topology T on X .2.3. Definition.
We call a point x ∈ ∗ X remote if x is not near-standard,i.e., not in the monad of any standard point of X . Given an equivalencerelation on the set of remote points of ∗ X , we write x ∼ y if x and y areremote and equivalent. Let Y be the point set consisting of points of X , called s-points , togetherwith all equivalence classes of remote points, where each such equivalenceclass is treated as a single point. We call each such point of Y an r-point .We supply ∗ X with the S-topology, that is the topology generated by thenonstandard extensions of standard open subsets of X . Let ϕ be the map-ping from ∗ X onto Y that sends near-standard points to their standard partsand remote points to their respective r-points. The neighborhood filterbase B ( y ) at an r-point y in Y consists of all sets of the form ϕ ( ∗ U ), where U is a standard open subset of X with nonstandard extension containing theentire equivalence class corresponding to y (whence U = ∅ ). The neigh-borhood filter base B ( x ) at an s-point x in Y consists of all sets of theform ϕ ( ∗ U ), where U is a standard open subset of X with x ∈ U . For eachpoint p ∈ Y , B ( p ) is in fact a filter base (see [3].) As usual, a set O ⊆ Y is called “open” if for each point p ∈ O , there is an element ϕ ( ∗ U ) ∈ B ( p )with ϕ ( ∗ U ) ⊆ O . The collection of open sets forms a topology on Y . Notethat we have not made any claim about the interior with respect to Y ofany member of any neighborhood filter base. We let T Y denote the topologyon Y , i.e., the collection of open sets, generated by the neighborhood filterbases. The following result is established in [3] using the fact (see [10] and[11]) that ∗ X with the S-topology is compact.2.4. Theorem.
The map ϕ is a continuous surjection from ∗ X onto Y ,whence, Y is compact. Moreover, the point set X is dense in Y suppliedwith the T Y -topology, and T is stronger than, or equal to, the relative T Y -topology on X . Recall that all sets in T are themselves in T Y if T is a locally compacttopology on X . The following is an example where the T -topology on X isstrictly stronger than the relative T Y -topology.2.5. Example.
Let X be the rational numbers in the interval [0 , x in ∗ X is remote if and only if x is in the monad of an irrational point in[0 , r -point, denoted by α , in Y . If U is a nonempty open set in X , then ∗ U contains remote points. In order for there to be a set V in T Y for whichthe restriction to X is U , it is necessary that V contains an element of theneighborhood filter base of α . That is, V must contain a set ϕ [ ∗ W ] where W is in T , and ∗ W contains every remote point in ∗ X . Such a subset W of X must contain every point of X except perhaps those in a compact subsetof X . Therefore, the point set Y consists of the points of X together with AUSDORFF COMPACTIFICATIONS 3 the point α , and the nonempty elements of T Y are the complements in Y ofcompact subsets of X .In the next section, we show that any Hausdorff compactification of aHausdorff space ( X, T ) can be produced with an equivalence relation onthe remote points of ∗ X . First, however, we give an example showing thatthis may not be true if the compactification is not Hausdorff. It is also anexample of a Hausdorff space ( X, T ) that is not locally compact, but stillforms an open set in a compactification.2.6. Example.
We modify Example 2 on Page 630 from [4]. Let X be thesubset of the plane given by X = { ( x,
0) : x rational, 0 ≤ x ≤ } . To form the compactification Y , we adjoint to X the subset ∆ in the planegiven by ∆ = { ( x,
1) : 0 ≤ x ≤ } . A typical neighborhood of a point ( x ,
0) in X is given by the relative planetopology. That is, it is given by a constant ε > { ( x, ∈ X : | x − x | < ε } . A typical neighborhood of a point ( x , ∈ ∆ is given by a constant δ > { ( x, ∈ ∆ : | x − x | < δ } ∪ { ( x, ∈ X : | x − x | < δ } . Clearly, X is dense in Y = X ∪ ∆, and Y is not Hausdorff. Moreover, Y is compact, since any net has a cluster point in ∆. The compactification Y cannot be obtained using an equivalence relation on the remote points of ∗ X when the neighborhoods are formed as in [3]. To see this, suppose ( r, ∗ X that is in the equivalence class forming the point(0 ,
1) in ∆. Then r is in the monad of a standard irrational s in [0 , U in X such that ( r,
0) is in ∗ U , and every near-standard point in ∗ U has x -coordinate less than s/
2. But this is impossible.3.
Hausdorff Compactifications
Assume that ( X, T ) is a Hausdorff space, and let ( Z, T Z ) be a Haus-dorff compactification of ( X, T ). That is, X is dense subset of the compactHausdorff space Z , and the mapping from X to X as a subset of Z is ahomeomorphism.3.1. Definition.
Remote points p and q in ∗ X are equivalent, i.e., p ∼ q , if p and q are in the monad of a point z ∈ Z \ X . Let ( Y, T Y ) be the compactspace produced with this equivalence relation. Denote by F the mapping from Z to Y that is the identity mapping from X as a subset of Z to X as a subsetof Y , and maps each point z ∈ Z \ X to the r-point in Y formed from theequivalence class formed by those points in ∗ X that are in the monad of z . MATT INSALL, PETER A. LOEB, AND MA LGORZATA ANETA MARCINIAK
Theorem.
The map F is a bijection, and indeed, a homeomorphismfrom ( Z, T Z ) onto ( Y, T Y ) . It follows that any Hausdorff compactification of X can be obtained from an equivalence relation on the remote points of ∗ X .Proof. It is clear that F is bijective. As is well known, it is sufficient toshow that F − is continuous. For then, the inverse image using F of anopen set in Y will be the open complement in Z of the compact image of aclosed, therefore compact set in Y . Fix z ∈ Z , and an open neighborhood U of z . By the regularity of ( Z, T Z ), there is an open neighborhood V of z for which the closure is contained in U . Let O = V ∩ X . Since ( Z, T Z ) is acompactification of ( X, T ), O ∈ T . Moreover, ∗ O contains F ( z ), and ϕ ( ∗ O )is a closed neighborhood of F ( z ) that maps, using F − , into U . It followsthat F − is continuous. (cid:3) Example.
As an application to moduli spaces, we compactify the spaceof similar triangles in R discussed by Madeline Brandt in [1]. Here, twotriangles are equivalent with respect to this space of triangles if they aresimilar. Each such equivalence class is represented by the member with aleg of shortest length on the interval in the y -axis from vertex (0 ,
0) to vertex(0 , S = (cid:16) ( x, y ) ∈ R : x > , y ≥ / , x + ( y − ≥ (cid:17) . Thinking of S as a subset of the complex plane, the remote points are thepoints in the nonstandard extension of S with argument in the monad of π/ ∗ (0 , π/ π/
2, and represent thatequivalence class with one point at 2 i . On the other hand, we can break upthat equivalence class so there is just one point for members with unlim-ited modulus, and represent that set with the nonnegative y -axis. We canthen call two remote points with infinitesimally close, limited moduli andargument in the monad of π/ S consisting of thestandard parts of those points of limited modulus in the triangle. We calltwo such remote points equivalent if their arguments have the same standardpart θ < π/
2. The corresponding triangles have the same standard part in S . The corresponding equivalence class can be represented by that standardpart, which is the figure formed by the line from (0 ,
0) to (0 ,
1) together withthe infinite line segment with endpoint (0 ,
0) forming an angle θ with the x -axis and the parallel infinite line segment with endpoint (0 , Example.
The Martin compactification (see [5]) is an important con-struction in potential theory; it can be seen as an application of Theorem3.14 in [3]. Martin’s boundary is also of great importance in probability
AUSDORFF COMPACTIFICATIONS 5 theory. Motivated by our Theorem 3.2, the second author obtained a prob-ability formulated equivalence relation in [6] that yields the Martin com-pactification for a number of illuminating examples. His approach, “lookinginside” a domain, only makes sense when one can speak of points that areneither points of an existing boundary nor points in a compact subset of thedomain.
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Top. Proc. (2012), 1–11.[3] M. Insall, P.A.Loeb, M. Marciniak, End compactifications and general compactifica-tions, Journal of Logic and Analysis (2014) 1–16.[4] P. A. Loeb, Compactifications of Hausdorff spaces,
Proc. Amer. Math. Soc. (1969),627–634.[5] J. Doob, Classical potential theory and its probabilistic counterpart , Springer, NewYork, Berlin, Heidelerg, 1984.[6] , An Intuitive Approach to the Martin Boundary,
Indagationes Mathematicae (2020), 879–884.[7] P. A. Loeb and M. Wolff, ed., Nonstandard Analysis for the Working Mathematician ,second edition, editors Loeb and Wolff, Springer, New York, Berlin, Heidelerg, 2015.[8] A. Robinson,
Non-standard Analysis , North-Holland, Amsterdam, 1966.[9] , Compactification of groups and rings and nonstandard analysis,
Jour. of Sym-bolic Logic (1969), 576–588.[10] S. Salbany and T. Todorov, Nonstandard Analysis in Topology: Nonstandard andStandard Compactifications, J. Symbolic Logic , (2000), 1836–1840.[11] , Lecture Notes: Nonstandard Analysis in Topology ,http://arxiv.org/abs/1107.3323.
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