By J. L. Dupont, I. H. Madsen
College of Aarhus, 50. Anniversary, eleven September 1978
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Extra info for Algebraic Topology, Aarhus 1978: Proceedings of a Symposium held at Aarhus, Denmark, August 7-12, 1978
7 For x, y, z ∈ , x ∨ y ∨ z = (x ∨ y) ∨ z = x ∨ (y ∨ z). 8 For a ≤ b in and x, y ∈ we obtain x ∨ a ≤ x ∨ b, a ∨ y ≤ b ∨ y. 9 Let id∧ ( ) be the set of ∧-idempotent elements of , that is, λ ∈ i ∧ ( ) if and only if λ ∧ λ = λ. For x ∈ id∧ ( ) and x ≤ z in we have x ∨ (x ∧ z) ≤ (x ∨ x) ∧ z, x ∨ (z ∧ x) ≤ (x ∨ z) ∧ x. 10 For x ∈ and λ1 , . , λn ∈ · · · ∨ (x ∧ λn ) = x. 10 as it has been formulated above only states that global covers induce covers on the left, that is, by taking ∧ from the left. 10 also has to be modified so that global covers induce covers via ∧ on the right.
1 Properties 1. From A1 , . , A4 , it follows that x ∧ y, y ∧ x ≤ x, y. 2. If x ∈ id∧ ( ) and x ≤ y, then x ∧ y = y ∧ x = x. 3. For x, y ∈ id∧ ( ), x ∧ y = y ∧ x yields x ∧ y ∈ id∧ ( ). Moreover, if x ∧ y and y ∧ x are in i ∧ ( ), then x ∧ y = y ∧ x. 4. 1, . 9, then id∧ ( ) ⊂ id∨ ( ). 5. For x ∈ id∧ ( ) and x ≤ z we obtain x ∨(y∧z) ≤ ((x ∨y)∧z)∨2 (x ∨(y∧z))∧2 ≤ (x ∨ y) ∧ z; where ∨2 and ∧2 are exponent notation with respect to ∨ and ∧, respectively. 6. For x, y, z ∈ with x ∈ id∧ ( ) we obtain x ∧ (y ∨ z) ≥ ((x ∧ y) ∨ (x ∧ z))∧2 ((x ∨ y) ∧ (x ∨ z))∨2 ≥ x ∨ (y ∧ z).
N for some ∧. -irreducible ρi , i = 1 . , n. Finally x = ρ1 ∧. . ∧. ρn with ρi ∧. -irreducible in id∧ ( ) follows. 1 mark only the beginning of a general theory; for purposes here, we do not need to develop this. Starting from a virtual topology say, where only the abstract opens are given, we may study closed sets in C( ); for example, we define the closure of [λ] in C( ), for λ ∈ , as follows: [A] ∈ [λ]cl , where A is directed in , if [µ] ≥ [A] with µ ∈ implies that µ∧λ = 0. It is clear that [λ] ∈ [λ]cl .