The Samuel realcompactification Ana S. Mero no Universidad - PowerPoint PPT Presentation

The Samuel realcompactification Ana S. Mero˜ no Universidad Complutense de Madrid Joint work with Prof. M. Isabel Garrido Ana S. Mero˜ no The Samuel realcompactification

Introduction. In this talk we will introduce a realcompactification for the class of uniform spaces ( X , µ ) and we will call it the Samuel realcompactification . Then we will study this realcompactification in the frame of metric spaces ( X , d ). First, we will compare the Samuel realcompactification of a metric space ( X , d ) with another type of realcompactification that can be defined for metric spaces and which is called Lipschitz realcompactification . Finally, we will see which metric spaces can be considered Samuel realcompact. Ana S. Mero˜ no The Samuel realcompactification

Contents. 1 General results about realcompactifications. 2 Examples of realcompactifications and compactifications. 3 Realcompactifications on metric spaces. 4 Equivalence of the Lipschitz and the Samuel realcompactification. 5 Samuel realcompact metric spaces. Ana S. Mero˜ no The Samuel realcompactification

Realcompactifications of a Tychonoff space. Definition. A realcompactification of a Tychonoff space X is a realcompact space Y in which X is densely embedded. compactification ⇒ realcompactification Ana S. Mero˜ no The Samuel realcompactification

Order in the realcompactifications. ( R ( X ), ≤ ) where R ( X ) = { realcompactifications of X } and ≤ is a partial order defined as follows: α 1 X ≤ α 2 X if there is h : α 2 X → α 1 X continuous, leaving X pointwise fixed Definition. Two realcompactifications α 1 X and α 2 X are equivalent whenever α 1 X ≤ α 2 X and α 2 X ≤ α 1 X . ∃ h : α 2 X → α 1 X an homeomorphism, leaving X pointwise fixed Ana S. Mero˜ no The Samuel realcompactification

Generation of realcompactifications. F ⊂ C ( X ) separating points from closed sets e : X → R F embedding e ( x ) = ( f ( x )) f ∈F R F H ( F ) = e ( X ) • H ( F ) is the smallest realcompactification of X such that every function f ∈ F can be continuously extended to it. • Whenever F has an algebraic structure, for instance, if F is a vector lattice, then H ( F ) = { real unital vector lattice homomorphisms on F} Ana S. Mero˜ no The Samuel realcompactification

Compactifications. F ∗ = F ∩ C ∗ ( X ) bounded functions of F • H ( F ∗ ) is the smallest compactification and realcompactification of X such that every function f ∈ F ∗ can be continuously extended to it. X ⊂ H ( F ) ⊂ H ( F ∗ ) Ana S. Mero˜ no The Samuel realcompactification

Hewitt realcompactification and Stone-ˇ Cech compactification. X Tychonoff space F = C ( X ) real-valued continuous functions H ( C ( X )) = υ X is the Hewitt realcompactification of X • υ X is largest element in the ordered family ( R ( X ) , ≤ ). • υ X is the smallest realcompactification of X such that every f ∈ C ( X ) is continuously extended. Ana S. Mero˜ no The Samuel realcompactification

Hewitt realcompactification and Stone-ˇ Cech compactification. F ∗ = C ∗ ( X ) bounded real-valued continuous functions H ( C ∗ ( X )) = β X is the Stone-ˇ Cech compactification of X • β X is the smallest compactification and realcompactification of X such that every f ∈ C ∗ ( X ) is continuously extended. X ⊂ υ X ⊂ β X Theorem. A Tychonoff space X is realcompact if and only if X = υ X . Ana S. Mero˜ no The Samuel realcompactification

Samuel realcompactification and compactification. ( X , µ ) uniform space F = U µ ( X ) real-valued uniformly continuous functions • H ( U µ ( X )) is the smallest realcompactification of X such that f ∈ U µ ( X ) is continuously extended. F ∗ = U ∗ µ ( X ) bounded real-valued uniformly continuous functions H ( U ∗ µ ( X )) = s µ X is the Samuel compactification of ( X , µ ) • s µ X is the smallest compactification and realcompactification of X such that every f ∈ U ∗ µ ( X ) is continuously extended. Ana S. Mero˜ no The Samuel realcompactification

Samuel realcompactification and compactification. X ⊂ H ( U µ ( X )) ⊂ s µ X We will call H ( U µ ( X )) the Samuel realcompactification of ( X , µ ) because it is associated to the family of all the real-valued uniformly continuous functions as the Samuel compactification is associated to the family of all the bounded real-valued uniformly continuous functions. Definition. A uniform space ( X , µ ) is Samuel realcompact if X = H ( U µ ( X )). Samuel realcompact ⇒ realcompact Ana S. Mero˜ no The Samuel realcompactification

Lipschitz realcompactification. ( X , d ) metric space F = Lip d ( X ) real-valued Lipschitz functions • H ( Lip d ( X )) is the smallest realcompactification of X such that every f ∈ Lip d ( X ) is continuously extended. We will call H ( Lip d ( X )) the Lipschitz realcompactification of ( X , d ) F ∗ = Lip ∗ d ( X ) bounded real-valued Lipschitz functions Theorem. H ( Lip ∗ d ( X )) is exactly the Samuel compactification s d X of ( X , d ) However in the unbounded case, H ( Lip d ( X )) and H ( U d ( X )) are in general different realcompactifications. Ana S. Mero˜ no The Samuel realcompactification

Lipschitz realcompactification. X ⊂ H ( Lip d ( X )) ⊂ s d ( X ) Definition. A metric space ( X , d ) is Lipschitz realcompact if X = H ( Lip d ( X )). Lipschitz realcompact ⇒ realcompact Ana S. Mero˜ no The Samuel realcompactification

Main reference. M. I. Garrido, A S. Mero˜ no , The Samuel realcompactification of a metric space (submitted) Ana S. Mero˜ no The Samuel realcompactification

Lipschitz realcompactification. Theorem. Let ( X , d ) be a metric space x 0 a fixed point in X and B d [ x 0 , n ] the closed ball of center x 0 and radius n ∈ N . Then � H ( Lip d ( X )) = cl s d X B d [ x 0 , n ] ⊂ s d ( X ) n ∈ N Corollary. A metric space is Lipschitz realcompact if and only if every closed bounded subset is compact. Ana S. Mero˜ no The Samuel realcompactification

Relations between the realcompactifications. X ⊂ υ X ⊂ H ( U d ( X )) ⊂ H ( Lip d ( X )) ⊂ s d X Lipschitz realcompact ⇒ Samuel realcompact u Observe that, uniformly equivalents metrics ρ ≃ d define identical Samuel realcompactifications and compactifications. u u � � � � � � H ( U d ( X )) = H ( Lip ρ ( X )) : ρ ≃ d = H ( Lip ρ ( X )) : ρ ≃ d . Ana S. Mero˜ no The Samuel realcompactification

Relations between the realcompactifications. t We write ρ ≃ d for topologically equivalent metrics. t t � � � � � � υ X = H ( U ρ ( X )) : ρ ≃ d = H ( U ρ ( X )) : ρ ≃ d t t � � � � � � υ X = H ( Lip ρ ( X )) : ρ ≃ d = H ( Lip ρ ( X )) : ρ ≃ d . Ana S. Mero˜ no The Samuel realcompactification

Problems. 1 To characterize those metric spaces ( X , d ) for which there exists a uniformly equivalent metric ρ such that H ( U d ( X )) and H ( Lip ρ ( X )) are equivalent realcompactifications, that is, H ( U d ( X )) = H ( Lip ρ ( X )). 2 To characterize those metric space ( X , d ) which are Samuel realcompact, that is, X = H ( U d ( X )). Ana S. Mero˜ no The Samuel realcompactification

Bourbaki-bounded subsets. B d ( x , ε ) be the open ball of center x ∈ X and radius ε > 0 B 2 � d ( x , ε ) = { B d ( y , ε ) : y ∈ B d ( x , ε ) } and � { B d ( y , ε ) : y ∈ B m − 1 B m d ( x , ε ) = ( x , ε ) } whenever m ≥ 3 . d Definition. A subset B of a metric space is Bourbaki-bounded if for every ε > 0 there exist finitely many points x 1 , ..., x k ∈ X such that for some m ∈ N , k � B m B ⊂ d ( x i , ε ) . i =1 Ana S. Mero˜ no The Samuel realcompactification

Bourbaki-bounded subsets. Theorem. (Atsuji) For a subset B of a metric space ( X , d ) the following statements are equivalent: 1 B is Bourbaki-bounded; 2 f ( B ) ⊂ R is bounded for every f ∈ U d ( X ). Ana S. Mero˜ no The Samuel realcompactification

Bourbaki-bounded subsets. Examples. 1 Totally bounded subsets of metric spaces are Bourbaki-bounded. 2 Bounded subsets of normed vector spaces are Bourbaki-bounded. 3 Let X = N × ℓ 2 where N brings the 0 − 1 discrete metric and ℓ 2 is the classical Hilbert spaces. Let d the product metric. Then, in ( X , d ), bounded subset are not Bourbaki-bounded and Bourbaki-bounded subsets are not totally bounded. Ana S. Mero˜ no The Samuel realcompactification

Hejcman’s problem. BB d ( X ) = { Bourbaki-bounded subsets } B d ( X ) = { bounded subsets } J. Hejcman , On simple recognizing of bounded sets Comment. Math. Univ. Carolinae, 38 (1997), 149-156. To determine those metric spaces ( X , d ) such that for some u uniformly equivalent ρ ≃ d, BB d ( X ) = B ρ ( X ) . Example. Every normed vector space satisfies that BB d ( X ) = B d ( X ). Ana S. Mero˜ no The Samuel realcompactification

Main result. Proposition For a metric space ( X , d ) the following statements are equivalent: u 1 there exists a uniformly equivalent metric ρ ≃ d such that H ( U d ( X )) is equivalent to H ( Lip ρ ( X )) ; 2 H ( U d ( X )) uniformly locally compact for the weak uniformity on H ( U d ( X )) as a uniform subspace of the product space R U d ( X ) ; 3 every uniform partition of X is countable and there exists ǫ > 0 such that for every x ∈ X and every m ∈ N , B m d ( x , ε ) is a Bourbaki-bounded subset; u 4 there exists a uniformly equivalent metric ρ ≃ d such that BB d ( X ) = B ρ ( X ) . Ana S. Mero˜ no The Samuel realcompactification