Topology from order Order from topology Interval topology Applications
Order and Topology
Andrew Craig
BOATS Meeting Birmingham 20 April 2010
Andrew Craig Order and Topology Topology from order Order from topology Interval topology Applications
Ordered sets
A partially ordered set (X , 6) is a set X with a relation 6 satisfying (P1) x 6 x (re exivity) (P2) x 6 y, y 6 z =⇒ x 6 z (transitivity) (P3) x 6 y, y 6 x =⇒ x = y (anti-symmetry) Also called ordered sets or posets. Pre-orders require only (P1) and (P2). Examples (℘(X ), ⊆)
(R, 6) (a linear order, or chain) n R with product order: (x1,..., xn) 6 (y1,..., yn) ⇐⇒ ∀i, xi 6 yi X R with point-wise order: f 6 g ⇐⇒ ∀x ∈ X , f (x) 6 g(x)
Andrew Craig Order and Topology Topology from order Order from topology Interval topology Applications
Morphisms
A map f :(X , 6X ) −→ (Y , 6Y ) is order-preserving if whenever a 6 b in X , then f (a) 6 f (b) in Y . Up-sets and down-sets For A ⊆ X ,
↑A = { x ∈ X : ∃y ∈ A, y 6 x } ↓A = { x ∈ X : ∃y ∈ A, x 6 y } A is an up-set if A = ↑A, a down-set if A = ↓A.
Complements of up-sets are down-sets and vice-versa.
We have the principal up-sets ↑x = { y : x 6 y } and principal down-sets ↓x = { y : y 6 x } for x ∈ X .
Andrew Craig Order and Topology Topology from order Order from topology Interval topology Applications
Order topologies
Alexandrov topology - A = { U : U = ↑U } Upper topology - υ subbasic opens of the form X \↓x Lower topology - ω subbasic opens of the form X \↑x Interval topology - ι = υ ∨ ω Also Scott topology - σ U ∈ σ ⇐⇒ U an up-set, for every D ⊆ X directed, if W D ∈ U, then U ∩ D 6= ∅ Lawson topology - λ = σ ∨ ω Scott topology important on dcpo's and in domain theory.
Andrew Craig Order and Topology Topology from order Order from topology Interval topology Applications
Specialization order
Let (X , τ ) be a topological space. De ne 6τ by:
x 6τ y ⇐⇒ x ∈ {y}
⇐⇒ Nx ⊆ Ny
Then (X , 6τ ) is a pre-order. If (X , τ ) is T0 =⇒ 6τ a partial order. If (X , τ ) is T1 =⇒ 6τ the discrete order. Question: For which order topologies will the specialization order agree with the original order?
Andrew Craig Order and Topology Topology from order Order from topology Interval topology Applications
Answer:
For any ordered set (X , 6), we have: υ ⊆ σ ⊆ A.
Further, and 6υ = 6 6A = 6 . In fact, 6τ = 6 ⇐⇒ υ ⊆ τ ⊆ A.
If X is nite, then υ = σ = A.
Andrew Craig Order and Topology Topology from order Order from topology Interval topology Applications
A topological space (X , τ ) is an Alexandrov space if the intersection of any collection of open sets is open. Also called nitely-generated spaces. Taking the specialization order, we get that V ⊆ X is open i it is an upper set wrt 6τ . We have two functors:
S : TOP −→ PRE, (X , τ ) 7−→ (X , 6τ )
A : PRE −→ TOP, (X , 6) 7−→ (X , A) If we restrict TOP to Alexandrov spaces then we have an isomorphism between the categories. f :(X , τ ) −→ (Y , δ) is continuous i it is order-preserving from X Y . ( , 6τ ) −→ ( , 6δ)
Andrew Craig Order and Topology Topology from order Order from topology Interval topology Applications
Interval topology Consider the interval topology on a linearly ordered set. This is a LOTS. Every LOTS is T0, T1,... T5. Question: Is the same true for partially ordered sets? Answer: No.
Lemma (Erné) (X , 6) is T2 i for x 6= y there exist nite sets F , G such that F ∩ {x, y}u 6= ∅ and G ∩ {x, y}l 6= ∅ and X = ↓F ∪ ↑G.
Andrew Craig Order and Topology Topology from order Order from topology Interval topology Applications
Theorem (Erné) If (X , 6) contains an in nite anti-chain then it is not T2. There exists an ordered set which is Hausdor but not regular.
Theorem (Wolk) A lattice is complete i it is compact in its interval topology.
Therefore, for a complete lattice, T2 ⇐⇒ regular ⇐⇒ normal.
Andrew Craig Order and Topology Topology from order Order from topology Interval topology Applications
Interval topology on products
For a family of ordered sets Xj , j ∈ J we have:
Y Y ι Xj ⊆ ιj (Xj ). j∈J j∈J Q Theorem (Erné) For a nite product X = j Xj of linearly ordered sets, the following are equivalent: Q (i) ι(X ) = j ι(Xj ); (ii) X has a greatest element, a least element, or ∃ k s.t. ∀ j 6= k, Xj bounded;
(iii) for all j 6= k, Xj has a greatest element, or Xk has a least element;
(iv) (X , ι) is a T2 space.
Andrew Craig Order and Topology Topology from order Order from topology Interval topology Applications
Example:
X1 × X2 where X1, X2 from [0, 1], (0, 1], [0, 1), (0, 1).
Further examples:
n For n > 1, the Euclidean topology on R is strictly ner than the interval topology.
n The interval topology on N is discrete. n The interval topology on R is distinct from the product topology.
Andrew Craig Order and Topology Topology from order Order from topology Interval topology Applications
Applications of order topologies in current research topic A lattice, L, is an ordered set such that for all x, y ∈ L: x ∨ y (least upper bound) and x ∧ y (greatest lower bound) exist.
A complete lattice has arbitrary meets and joins. i.e. for any A ⊆ L, W A ∈ L and V A ∈ L.
A completion is an embedding of a lattice L into a complete lattice C:
L ,→ C
Andrew Craig Order and Topology Topology from order Order from topology Interval topology Applications
Topology can be used to extend maps between lattices to maps between completions. Let L, M be lattices and C a completion of L. Let τ be a topology on C.
Suppose f : L −→ M and e : L ,→ C. For x ∈ C: _ n ^ o f σ(x) = f (U ∩ L): x ∈ U ∈ τ
^ n _ o f π(x) = f (U ∩ L): x ∈ U ∈ τ
σ π σ π Both f and f extend f , and f 6 f .
Andrew Craig Order and Topology Topology from order Order from topology Interval topology Applications
References
Introduction to Lattices and Order, Davey and Priestley Continuous Lattices and Domains, Gierz et al. The ABC of order and topology, M. Erné, in Category theory at work Separation axioms for interval topologies, M. Erné, 1980 Topologies on products of partially ordered sets I, M. Erné, 1980
Andrew Craig Order and Topology