Order properties of bases in products David Milovich Texas A&M - PowerPoint PPT Presentation

Order properties of bases in products David Milovich Texas A&M International University http://www.tamiu.edu/ ∼ dmilovich david.milovich@tamiu.edu Joint work with Guit-Jan Ridderbos and Santi Spadaro Mar. 20, 2010 Spring Topology and Dynamics Conference Mississippi State University

Order theory preliminaries Definition ◮ A preorder P is κ -directed if every subset smaller than κ has an (upper) bound in P . ◮ Directed means ℵ 0 -directed.

Order theory preliminaries Definition ◮ A preorder P is κ -directed if every subset smaller than κ has an (upper) bound in P . ◮ Directed means ℵ 0 -directed. Conversely: ◮ A preorder P is κ -founded if every bounded subset is smaller than κ . ◮ Flat means ℵ 0 -founded.

Order theory preliminaries Definition ◮ A preorder P is κ -directed if every subset smaller than κ has an (upper) bound in P . ◮ Directed means ℵ 0 -directed. Conversely: ◮ A preorder P is κ -founded if every bounded subset is smaller than κ . ◮ Flat means ℵ 0 -founded. Definition A preorder P is almost κ -founded if it has a κ -founded cofinal suborder.

Order theory preliminaries Definition ◮ A preorder P is κ -directed if every subset smaller than κ has an (upper) bound in P . ◮ Directed means ℵ 0 -directed. Conversely: ◮ A preorder P is κ -founded if every bounded subset is smaller than κ . ◮ Flat means ℵ 0 -founded. Definition A preorder P is almost κ -founded if it has a κ -founded cofinal suborder. Convention Order sets like κ , [ λ ] κ , and 2 <κ by ⊆ .

Topological preliminaries Convention ◮ All spaces are Hausdorff ( T 2 ). ◮ Families of open sets are ordered by ⊇ .

Topological preliminaries Convention ◮ All spaces are Hausdorff ( T 2 ). ◮ Families of open sets are ordered by ⊇ . Notation ◮ τ ( X ) is the set of open subsets of X . ◮ τ + ( X ) is the set of nonempty open subsets of X ◮ τ ( p , X ) is the set of open neighborhoods of p in X .

Topological preliminaries Convention ◮ All spaces are Hausdorff ( T 2 ). ◮ Families of open sets are ordered by ⊇ . Notation ◮ τ ( X ) is the set of open subsets of X . ◮ τ + ( X ) is the set of nonempty open subsets of X ◮ τ ( p , X ) is the set of open neighborhoods of p in X . Definition ◮ A local base at p is a cofinal subset of τ ( p , X ). ◮ A π -base is a cofinal subset of τ + ( X ). ◮ A base is a subset B of τ ( X ) that includes a local base at every point.

The weight The Noetherian type w ( X ) of X is Nt ( X ) of X is the least κ ≥ ℵ 0 such that the least κ ≥ ℵ 0 such that X has a base that is X has a base that is of size ≤ κ . κ -founded. The π -weight The Noetherian π -type π ( X ) of X is π Nt ( X ) of X is the least κ ≥ ℵ 0 such that the least κ ≥ ℵ 0 such that X has a π -base that is X has a π -base that is of size ≤ κ . κ -founded. The character The local Noetherian type χ ( p , X ) of p in X is χ Nt ( p , X ) of p in X is the least κ ≥ ℵ 0 such that the least κ ≥ ℵ 0 such that p has a local base that is p has a local base that is of size ≤ κ . κ -founded. χ ( X ) = sup p ∈ X χ ( p , X ) χ Nt ( X ) = sup p ∈ X χ Nt ( p , X )

History ◮ Malykhin, Peregudov, and ˇ Sapirovski˘ i studied the properties Nt ( X ) ≤ ℵ 1 , π Nt ( X ) ≤ ℵ 1 , Nt ( X ) = ℵ 0 , and π Nt ( X ) = ℵ 0 in the 1970s and 1980s. ◮ Peregudov introduced Noetherian type and Noetherian π -type in 1997. ◮ Bennett and Lutzer rediscovered the property Nt ( X ) = ℵ 0 in 1998. ◮ In 2005, Milovich introduced local Noetherian type and rediscovered Noetherian type and Noetherian π -type.

Easy upper bounds Lemma Every preorder P is almost cf( P )-founded. Corollary For all spaces X , ◮ χ Nt ( p , X ) ≤ χ ( p , X ); ◮ χ Nt ( X ) ≤ χ ( X ); ◮ π Nt ( X ) ≤ π ( X ).

Easy upper bounds Lemma Every preorder P is almost cf( P )-founded. Corollary For all spaces X , ◮ χ Nt ( p , X ) ≤ χ ( p , X ); ◮ χ Nt ( X ) ≤ χ ( X ); ◮ π Nt ( X ) ≤ π ( X ). Even easier: Every P is | P | + -founded, so Nt ( X ) ≤ w ( X ) + .

Easy upper bounds Lemma Every preorder P is almost cf( P )-founded. Corollary For all spaces X , ◮ χ Nt ( p , X ) ≤ χ ( p , X ); ◮ χ Nt ( X ) ≤ χ ( X ); ◮ π Nt ( X ) ≤ π ( X ). Even easier: Every P is | P | + -founded, so Nt ( X ) ≤ w ( X ) + . Example Nt ( β N ) = w ( β N ) + = c + because π ( β N ) = ℵ 0 < cf( w ( β N )).

Easy upper bounds for products Theorem If p ∈ X = � i ∈ I X i , then: ◮ Nt ( X ) ≤ sup i ∈ I Nt ( X i ) (Peregudov, 1997) ◮ π Nt ( X ) ≤ sup i ∈ I π Nt ( X i ) ◮ χ Nt ( p , X ) ≤ sup i ∈ I χ Nt ( p ( i ) , X i ) ◮ χ Nt ( X ) ≤ sup i ∈ I χ Nt ( X )

Large products Theorem (essentially (Malykhin, 1981)) If X = � α<κ X α and | X α | > 1 for all α < κ , then ◮ κ ≥ χ ( p , X ) ⇒ χ Nt ( p , X ) = ℵ 0 ; ◮ κ ≥ χ ( X ) ⇒ χ Nt ( X ) = ℵ 0 ; ◮ κ ≥ π ( X ) ⇒ π Nt ( X ) = ℵ 0 ; ◮ κ ≥ w ( X ) ⇒ Nt ( X ) = ℵ 0 .

Corollary X × 2 w ( X ) � ◮ Nt � = ℵ 0 . (Malykhin, 1981)

Corollary X × 2 w ( X ) � ◮ Nt � = ℵ 0 . (Malykhin, 1981) � X × 2 π ( X ) � ◮ π Nt = ℵ 0 . X × 2 χ ( X ) � ◮ χ Nt � = ℵ 0 .

Corollary X × 2 w ( X ) � ◮ Nt � = ℵ 0 . (Malykhin, 1981) � X × 2 π ( X ) � ◮ π Nt = ℵ 0 . X × 2 χ ( X ) � ◮ χ Nt � = ℵ 0 . X w ( X ) � ◮ Nt � = ℵ 0 . X π ( X ) � ◮ π Nt � = ℵ 0 . � X χ ( X ) � ◮ χ Nt = ℵ 0 .

Finite powers Definition ◮ In a product space X = � i ∈ I X i , let Nt box ( X ) denote the least κ for which X has κ -founded base ( π -base, local base at p ) that consists only of boxes. ◮ Similarlly define χ Nt box ( p , X ). ◮ χ Nt box ( p , X ) = sup p ∈ X χ Nt box ( p , X ).

Finite powers Definition ◮ In a product space X = � i ∈ I X i , let Nt box ( X ) denote the least κ for which X has κ -founded base ( π -base, local base at p ) that consists only of boxes. ◮ Similarlly define χ Nt box ( p , X ). ◮ χ Nt box ( p , X ) = sup p ∈ X χ Nt box ( p , X ). Theorem (M.) For all n ∈ [1 , ω ), for all spaces X : χ Nt ( p n , X n ) = χ Nt box ( p n , X n ) = χ Nt ( p , X ) χ Nt ( X n ) = χ Nt box ( X n ) = χ Nt ( X ) Nt box ( X n ) = Nt ( X )

Could Nt ( X n ) � = Nt box ( X n )? Passing to subsets ◮ If B is a local base at p in X , then B includes a χ Nt ( X )-founded local base at p in X .

Could Nt ( X n ) � = Nt box ( X n )? Passing to subsets ◮ If B is a local base at p in X , then B includes a χ Nt ( X )-founded local base at p in X . ◮ If B is a π -base of X , then B includes a π Nt ( X )-founded π -base of X .

Could Nt ( X n ) � = Nt box ( X n )? Passing to subsets ◮ If B is a local base at p in X , then B includes a χ Nt ( X )-founded local base at p in X . ◮ If B is a π -base of X , then B includes a π Nt ( X )-founded π -base of X . ◮ The analogous claim for bases is false.

Could Nt ( X n ) � = Nt box ( X n )? Passing to subsets ◮ If B is a local base at p in X , then B includes a χ Nt ( X )-founded local base at p in X . ◮ If B is a π -base of X , then B includes a π Nt ( X )-founded π -base of X . ◮ The analogous claim for bases is false. Theorem (Bennett, Lutzer, 1998) Every metrizable space has a flat base. Proof : For each n < ω , pick a locally finite open cover refining the balls of radius 2 − n . Take the union of these covers.

Could Nt ( X n ) � = Nt box ( X n )? Passing to subsets ◮ If B is a local base at p in X , then B includes a χ Nt ( X )-founded local base at p in X . ◮ If B is a π -base of X , then B includes a π Nt ( X )-founded π -base of X . ◮ The analogous claim for bases is false. Theorem (Bennett, Lutzer, 1998) Every metrizable space has a flat base. Proof : For each n < ω , pick a locally finite open cover refining the balls of radius 2 − n . Take the union of these covers. Example (M., 2009) Set X = ω ω . Let B be the set of all sets of the form U s , n where s ∈ ω <ω , n < ω , and U s , n is the set of all f ∈ X such that s ⌢ i ⊆ f for some i ≤ n . B a base of X , but B has no flat subcover.

The Square Problem Open Question � X 2 � � = Nt ( X ) possible? (Recall Nt ( X ) = Nt box ( X 2 ).) Is Nt

The Square Problem Open Question � X 2 � � = Nt ( X ) possible? (Recall Nt ( X ) = Nt box ( X 2 ).) Is Nt (Balogh, Bennett, Burke, Gruenhage, Lutzer, and Mashburn � X 2 � (2001) asked if Nt � = Nt ( X ) = ℵ 0 is possible.)

The Square Problem Open Question � X 2 � � = Nt ( X ) possible? (Recall Nt ( X ) = Nt box ( X 2 ).) Is Nt (Balogh, Bennett, Burke, Gruenhage, Lutzer, and Mashburn � X 2 � (2001) asked if Nt � = Nt ( X ) = ℵ 0 is possible.) Partial answers (M., Spadaro) “No,” if: ◮ X is locally compact and metrizable;

The Square Problem Open Question � X 2 � � = Nt ( X ) possible? (Recall Nt ( X ) = Nt box ( X 2 ).) Is Nt (Balogh, Bennett, Burke, Gruenhage, Lutzer, and Mashburn � X 2 � (2001) asked if Nt � = Nt ( X ) = ℵ 0 is possible.) Partial answers (M., Spadaro) “No,” if: ◮ X is locally compact and metrizable; ◮ X is σ -compact and metrizable;