Proximity inductive dimension and Brouwer dimension agree on compact Hausdorff spaces
aa r X i v : . [ m a t h . GN ] M a r PROXIMITY INDUCTIVE DIMENSION AND BROUWERDIMENSION AGREE ON COMPACT HAUSDORFF SPACES
JEREMY SIEGERT
Abstract.
In this paper we show that the proximity inductive dimensiondefined by Isbell agrees with the Brouwer dimension originally described byBrouwer on the class of compact Hausdorff spaces. Consequently, Fedorchuk’sexample of a compact Hausdorff space whose Brouwer dimension exceeds itsLebesgue covering dimension is an example of a space whose proximity induc-tive dimension exceeds its proximity dimension as defined by Smirnov. Introduction
Proximity spaces in their modern form were described during the early 1950’sby Efremoviˇc, [1],[2]. Variations of the original definition can be found in [9]. Thestructure is meant to capture the notion of what it means for two subsets of a spaceto be “close”. Every proximity space has a natural completely regular topologicalstructure and the class of proximity spaces is placed somewhat neatly betweentopological spaces and uniform spaces in that every uniform space induces a prox-imity structure whose corresponding topology is the uniform topology. Likewiseevery proximity space is induced by at least one uniform structure (see [9]). Thedimension theory of proximity spaces began when Smirnov defined the proximitydimension δd of proximity spaces using δ -coverings, [10]. This dimension functionserves as a proximity invariant analog of the covering dimension dim . In the caseof compact Hausdorff spaces, whose topology is induced by a unique proximity, thedimensions δd and dim coincide. A proximity invariant inductive dimension wouldnot be defined until Isbell defined the notion of a “freeing set” and subsequentlythe proximity inductive dimension δInd in [5]. Isbell remarked in [6] and [7] thathe did not know of an instance where δInd and δd did not agree. In this paperwe will show that a space constructed by Fedorchuk in [4] has distinct δInd and δd . We do this by shown that δInd and the Brouwer dimension Dg agree on theclass of compact Hausdorff spaces. For the sake of self-containment we review thenecessary preliminary definitions in sections 2 and 3 before proceeding to our mainresults in section 4. Throughout this paper we use the notation A and int ( A ) forthe closure and interior of a subset A within a topological space X . Date : March 6, 2019.2010
Mathematics Subject Classification.
Key words and phrases. proximity, Brouwer dimension, dimensiongrad, inductive dimension. Proximity Spaces and their dimensions
In this section we will review the necessary definitions and results surroundingproximity spaces. The citations are not necessarily where the corresponding defi-nitions or results first appeared, but where they can be easily found. These initialdefinitions and results about proximity spaces can be found in [9].
Definition 2.1.
Let X be a set and δ a binary relation on 2 X . The relation δ issaid to be a proximity relation , or simply a proximity on X , if the followingaxioms are satisfied for all A, B, C ⊆ X :(1) AδB if and only if
BδA .(2) ( A ∪ B ) δC if and only if AδC or BδC .(3)
AδB implies that A = ∅ and B = ∅ .(4) A ∩ B = ∅ implies that AδB .(5) A ¯ δB implies that there is an E ⊆ X such that A ¯ δE and ( X \ E )¯ δB .Where A ¯ δB is interpreted as “ AδB is not true”. A pair (
X, δ ) where X is a setand δ is a proximity on X is called a proximity space . If for a proximity space( X, δ ) the relation δ satisfies the additional axiom that for all x, y ∈ X , { x } δ { y } if and only if x = y we call the proximity ( X, δ ) separated .As mentioned in the introduction every proximity space has a natural topologicalstructure. This topology is defined in the following way: Proposition 2.2. If ( X, δ ) is a proximity space then the function cl : 2 X → X defined by cl ( A ) := { x ∈ X | { x } δA } is a closure operator on X . Moreover, the corresponding topology is Hausdorffif and only if ( X, δ ) is separated. We will call the topology on a proximity space (
X, δ ) described above the topol-ogy induced by the proximity δ . A set U ⊆ X is open in the induced topologyif and only if for all x ∈ U one has that { x } ¯ δ ( X \ U ). Proposition 2.3.
Let ( X, δ ) be a proximity space, then for all A, B ⊆ XAδB ⇐⇒ AδB where A and B denote closure within the topology induced by δ . Definition 2.4.
Given a topological space X , a proximity relation δ on X is saidto be compatible with the topology on X if the topology induced by δ is theoriginal topology on X . Proposition 2.5. If X is a compact Hausdorff space, then there is a unique prox-imity on X that is compatible with the topology on X . It is defined by: ROXIMITY INDUCTIVE DIMENSION AND BROUWER DIMENSION 3
AδB ⇐⇒ A ∩ B = ∅ where A denotes the closure of A in X . Definition 2.6. If A and B are subsets of a proximity space ( X, δ ) then we saythat B is a δ - neighbourhood of A if A ¯ δ ( X \ B ). We denote this relationship bywriting A ≪ B .An elementary consequence of axiom (5) of Definition 2.1 is that if A and B are subsets of a proximity space ( X, δ ) such that A ≪ B then there is a C ⊆ X such that A ≪ C ≪ B . Moreover, it is an easy exercise to show that A ≪ B ifand only if A ≪ B . That is, a set and its closure in the induced topology havethe same δ -neighbourhoods. A useful fact about δ -neighbourhoods in proximityspaces whose topology is T is the following: Lemma 2.7.
Let ( X, δ ) be a separated proximity space whose induced topology is T . If A, B ⊆ X are such that and A ≪ B then A ⊆ int ( B ) where int ( B ) denotesthe interior of B in X .Proof. If A ≪ B then A ≪ B . By definition we then have that A ¯ δ ( X \ B ), whichimplies that A ¯ δ ( X \ B ). Then A and ( X \ B ) are disjoint closed subsets of X .Because X is T there is an open set U ⊆ X such that A ⊆ U and U ∩ ( X \ B ) = ∅ .Therefore A ⊆ U ⊆ B , which is to say that A ⊆ int ( B ). (cid:3) With these initial basic definitions and results in hand we will proceed to statingthe definitions and important results surround the proximity dimension δd . Thesedefinitions and results can be found in [10]. Definition 2.8.
Given a proximity space (
X, δ ) a finite cover A , . . . , A n of X iscalled a δ -cover if there is another finite cover B , . . . , B n of X such that B i ≪ A i for i = 1 , . . . , n . Definition 2.9.
Given a proximity space (
X, δ ) the proximity dimension of X ,denoted δd ( X ), is defined in the following way:(1) δd ( X ) = − X = ∅ .(2) For n ≥ δd ( X ) ≤ n if and only if every δ -cover U can be refined by a δ -cover of order at most n + 1.(3) δd ( X ) is the least integer n such that δd ( X ) ≤ n . If there is no such integerthen δd ( X ) = ∞ . Theorem 2.10. If X is a compact hausdorff space, then δd ( X ) = dim ( X ) . Note that there is no ambiguity in Theorem 2.10 as Proposition 2.5 grants thatthere is only one possible interpretation of δd on compact Hausdorff spaces. JEREMY SIEGERT
Next we proceed to Isbell’s proximity inductive dimension. These definitionsand results can be found in [7].
Definition 2.11.
Given a proximity space (
X, δ ) and two subsets
A, B ⊆ X suchthat A ¯ δB , a subset D ⊆ X is said to δ - separate A and B , or is a δ - separator between A and B , if X \ D = U ∪ V where A ⊆ U, B ⊆ V, U ∩ V = ∅ , and U ¯ δV . A subset H ⊆ X is said to free A and B , or be a freeing set for A and B , if H ¯ δ ( A ∪ B ) and every δ -neighbourhood of H that is disjoint from A ∪ B is a δ -separator between A and B . Definition 2.12.
Given a proximity space (
X, δ ) the proximity inductive di-mension of X , denoted δInd ( X ), is defined inductively:(1) δInd ( X ) = − X = ∅ .(2) For n ≥ δInd ( X ) ≤ n if and only if for every pair of subsets A, B ⊆ X such that A ¯ δB there is a set H ⊆ X that frees A and B and is such that δInd ( H ) ≤ n − δInd ( X ) is the least integer n such that δInd ( X ) ≤ n . If there is no such n , then δInd ( X ) = ∞ . Proposition 2.13. If ( Y, δ ) is a proximity space and X ⊆ Y is a dense subspace,then δInd ( X ) ≥ δInd ( Y ) . Proposition 2.14.
For every proximity space ( X, δ ) , δInd ( X ) ≥ δd ( X ) . We note that Proposition 2.13 implies that in Definition 2.12 we could havetaken
A, B, and H to be closed without altering the value of δInd . Definition 2.15.
Let X be a topological space and A, B ⊆ X disjoint closedsubsets. A closed set C ⊆ X is called a separator in X between A and B if thereare disjoint open sets U, V ⊆ X such that X \ C = U ∪ V , with A ⊆ U and V ⊆ V .The following result is an easy exercise, whose proof can be found in [6]. Proposition 2.16.
Let X be a compact Hausdorff space and A, B ⊆ X disjointclosed subsets. If C ⊆ X is a separator in X between A and B , then C frees A and B . The converse to the above result is not true.
Example 2.17.
Let X be the “Topologist’s Sine Curve”. That is X is the closureof the set A = { ( x, sin(1 /x )) ∈ R | x ∈ (0 , } in R . If we define A = { (0 , − } , B = { (1 , sin(1)) } , and C = { (0 , } then C frees A and B , but is not a separator between them. ROXIMITY INDUCTIVE DIMENSION AND BROUWER DIMENSION 5 Brouwer Dimension
Here we will review the basic definitions and results surround the Brouwer di-mension. For a brief history of the invariant see [3] or [4].
Definition 3.1. A continuum is a nonempty compact connected Hausdorff space.In some places in the literature (such as [8]) the word “compactum” is used fornonempty compact connected Hausdorff spaces. This is likely to distinguish themore general definition from the perhaps more standard definition of a continuumas a nonempty compact connected metric space. Definition 3.2.
Given a topological space X and two disjoint closed subsets A, B ⊆ X , a closed subset C ⊆ X that is disjoint from ( A ∪ B ) is called a cut between A and B if every continuum K ⊆ X such that K ∩ A = ∅ and K ∩ B = ∅ also satisfies K ∩ C = ∅ .It is an easy exercise to show that every separator in a topological space is alsoa cut. However as Example 2.17 shows, not every cut is a separator. Definition 3.3.
Let X be a T topological space. The Brouwer dimension of X , denoted Dg ( X ), is defined inductively:(1) Dg ( X ) = − X = ∅ .(2) For n ≥ Dg ( X ) ≤ n if and only if for every pair of disjoint closed sets A, B ⊆ X there is a cut C ⊆ X between A and B such that Dg ( C ) ≤ n − Dg ( X ) is the least integer n such that Dg ( X ) ≤ n . If there is no suchinteger then Dg ( X ) = ∞ .The following result appears in [8] and will be used in the next section. Lemma 3.4.
Let X be a compact Hausdorff space, A and B disjoint closed subsetsof X . If there is no connected set K such that K ∩ A = ∅ and K ∩ B = ∅ , thenthe empty set separates A and B . Said differently, this Lemma states that if the empty set is a cut between disjointclosed subsets of a compact Hausdorff space, then it is also a separator betweenthem. 4.
Main Results
In this final section we will prove that the proximity inductive dimension andthe Brouwer dimension coincide on compact Hausdorff spaces. To do this we firstcharacterize cuts within compact Hausdorff spaces.
Proposition 4.1.
Let X be a compact Hausdorff space and A, B ⊆ X nonemptydisjoint closed subsets. Then given a closed subset C ⊆ X that is disjoint from A ∪ B , the following are equivalent:(1) C is a cut in X between A and B . JEREMY SIEGERT (2) Every closed neighbourhood of C that is disjoint from A ∪ B is a separatorbetween A and B .Proof. Let
X, A, B, and C be given. We may assume without loss of generalitythat A and B are nonempty. Otherwise every closed set disjoint from A ∪ B is acut between A and B .((2) = ⇒ (1)) Assume that every closed neighbourhood D of C that is dis-joint from A ∪ B is a separator between A and B . If C = ∅ then C is a closedneighbourhood of itself that is disjoint from A and B , which would imply thatthe empty set is a separator between A and B , which would imply that C is acut between A and B . Assume then that C = ∅ and assume further towards acontradiction that C is not a cut. Then there is a continuum K ⊆ X such that K ∩ A = ∅ , K ∩ B = ∅ , but K ∩ C = ∅ . As K and C are disjoint closed sets there isa closed neighbourhood D of C that is disjoint from K . Then K ⊆ X \ D = U ∪ V where U and V are disjoint open sets of X containing A and B respectively. Thiswould imply that K ∩ U and K ∩ V are open subsets of K that witness K beingdisconnected, contradicting the connectedness of K . Therefore C is a cut between A and B .((1) = ⇒ (2)) Now assume that C is a cut between A and B . Let D be aclosed neighbourhood of C that is disjoint from A and B . Then C ⊆ int ( D ) andbecause C is a cut between A and B we have that there is no connected set K inthe compact Hausdorff space X \ int ( D ) that intersects both A and B nontrivially.Therefore by Lemma 3.4 the empty set is a separator in X \ int ( D ) between A and B . Then let U and V be disjoint open subset of X \ int ( D ) that contain A and B , respectively. Then U ′ := U ∩ ( X \ D ) and V ′ := V ∩ ( X \ D ) aredisjoint open subsets of X \ D and consequently disjoint open subsets of X suchthat X \ D = U ′ ∪ V ′ , A ⊆ U ′ , and B ⊆ V ′ . Therefore D is a separator in X between A and B . (cid:3) Proposition 4.2.
Let X be a compact Hausdorff space, and A, B ⊆ X disjointclosed sets. Given a closed subset C ⊆ X that is disjoint from A and B , thefollowing are equivalent:(1) C is a cut between A and B .(2) C frees A and B .Proof. Let
X, A, B, and C be given. As before we may assume without loss ofgenerality that A, B, and C are nonempty as the result is trivial otherwise.((2) = ⇒ (1)) Assume that C frees A and B . Every closed neighbourhood D of C that is disjoint from A and B is a δ -neighbourhood of C . Then by the definitionof a freeing set we have that X \ D = U ∪ V where A ⊆ U , B ⊆ V , and U ¯ δV .Then U ¯ δV in the subspace X \ D . Because U ¯ δV implies that U and V are disjoint ROXIMITY INDUCTIVE DIMENSION AND BROUWER DIMENSION 7 we then have that U and V are open in X \ D , and are therefore open in X . Wethen in fact have that D is a separator between A and B . As D was arbitrary wethen have that C is a cut between A and B by Proposition 4.1.((1) = ⇒ (2)) Assume that C is a cut between A and B and let D ⊆ X be a δ -neighbourhood of C that is disjoint from A ∪ B . We may assume that D is anopen neighbourhood of C because if C ≪ D then C ⊆ int ( D ) by Lemma 2.7 andif a subset of D is a δ -separator then D is as well. We then let D ′ be a closedneighbourhood of C such that C ⊆ D ′ ⊆ D and D ′ ∩ ( X \ D ) = ∅ . Then D ′ is aclosed neighbourhood of C that is disjoint from A and B , so by Proposition 4.1 D ′ is a separator in X between A and B . Then let U and V be disjoint open subsetsof X so that X \ D ′ = U ∪ V , A ⊆ U , and B ⊆ V . We then consider U ′ = U ∩ ( X \ D ) and V ′ := V ∩ ( X \ D )and claim that U ′ ¯ δV ′ . To see this we note that because U ′ and V ′ are subsets ofthe closed set X \ D we must have that U ′ and V ′ are also subsets of X \ D . Then U ′ ∩ V ′ ⊆ X \ D . However, as U ′ ⊆ U and V ′ ⊆ V we must have that U ′ ⊆ U and V ′ ⊆ V . Therefore we have U ′ ∩ V ′ ⊆ X \ D and U ′ ∩ V ′ ⊆ U ∩ V However, U ∩ V is a subset of D ′ and D ′ ∩ ( X \ D ) = ∅ . Therefore we have that U ′ ∩ V ′ = ∅ , which gives us that U ′ ¯ δV ′ . In summary, X \ D is the union of thedisjoint sets U ′ and V ′ that contain A and B respectively, and are not close. Thatis, D is a δ -separator between A and B , which implies that C frees A and B . (cid:3) Theorem 4.3.
For every compact Hausdorff space, Dg ( X ) = δInd ( X ) .Proof. We will show that δInd ( X ) ≥ Dg ( X ) by induction on δInd ( X ). Theresult is obvious when δInd ( X ) = −
1. Assume then that the result holds for δInd ( X ) < n for n ≥ δInd ( X ) = n . If A and B are disjointclosed subsets of X then there must be a closed set C ⊆ X that frees A and B and satisfies δInd ( C ) ≤ n −
1. By Proposition 4.2 C is a cut between A and B ,and the inductive hypothesis gives us that Dg ( C ) ≤ δInd ( C ) ≤ n −
1. Therefore Dg ( X ) ≤ n . Clearly, if δInd ( X ) = ∞ then Dg ( X ) ≤ δInd ( X ). Therefore δInd ( X ) ≥ Dg ( X ). The argument showing that Dg ( X ) ≥ δInd ( X ) is a similarinduction argument. (cid:3) In [4] a compact Hausdorff space B was constructed with the property that Dg ( X ) = 3 and dim ( X ) = 2. Then the conjunction of Theorem 4.3 and Theorem2.10 grants us the following corollary. Corollary 4.4.
There is a compact Hausdorff space B such that δInd ( B ) = 3 and δd ( B ) = 2 . JEREMY SIEGERT
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