77
Topology and Logic Programming
lim i s i ≡ s (C) if and only if for each x ∈ X eventually s i (x) = s(x)
determines a convergence class on [X → Y ] whose associated topology Q is
the product of X copies of the discrete topology on Y .
Proof: We must verify that the conditions (1), (2), (3), and (4) in the definition of a convergence class, see Definition 3.1.2, hold with the given meaning
of lim i s i ≡ s (C).
(1) Suppose that s i = s for all i ∈ I is a constant net. Then s i (x) = s(x)
for all x and all i. Hence, for all x, eventually s i (x) = s(x), and so ((s i ), s) ∈ C.
(2) Suppose that ((s i ), s) ∈ C and that (t j ) j
is a subnet of (s i ) i . Let
∈J
∈I
x ∈ X be arbitrary, and let i 0 be such that s i (x) = s(x) for all i ≥ i 0 . Since
(t j ) is a subnet of (s i ), there is φ : J → I and j 0 ∈ J such that i 0 ≤ φ(j)
whenever j 0 ≤ j. But then, if j 0 ≤ j, we have t j (x) = s φ(j) (x) = s(x), and
hence ((t j ), s) ∈ C.
(3) Suppose that (s i ) i does not converge (C) to s. Then there is x ∈ X
∈I
and a cofinal subset J of I such that, whenever j ∈ J , we have s j (x) = s(x).
Let t j = s j for each j ∈ J . Then (t j ) is a subnet of (s i ), and clearly no subnet
of (t j ) converges (C) to s.
(4) Suppose that the conditions stated in (4) of Definition 3.1.2 all hold
and that lim m lim n x(m, n) ≡ s (C), where x : F
' → [X → Y ]. Consider the
net x ◦ r : F → [X → Y ]. Let y ∈ X be arbitrary. Since lim m lim n x(m, n) ≡
s (C), there is m 0 ∈ I such that, for all m ≥ m 0 , lim n x(m, n) ≡ s m (C)
for some s m ∈ [X → Y ], and lim m s m ≡ s (C). Therefore, for m ≥ m 0 ,
there is n m ∈ J m such that x(m, n)(
s y) = s m (y) for all n ≥ n m . But for
m ≥ m 0 , s m (y) = s(y). Define f ∈ m I J m by setting f (m) = n m ∈ J
∈
m
whenever m ≥ m 0 and otherwise letting f (m) ∈ J m be arbitrary. Suppose
(m, g) ≥ (m 0 , f ). Then m ≥ m 0 and g ≥ f so that g(m) ≥ f (m) = n m . But
then we have x(m, g(m))(y) = s m (y) = s(y). In other words, (x◦r)(m, g)(y) =
x(m, g(m))(y) = s(y) whenever (m, g) ≥ (m 0 , f ). Thus, (x◦r)(y) is eventually
equal to s(y), and hence x ◦ r converges (C) to s.
Finally, viewing [X → Y ] as the product
s
x
Y , where Y = Y for each
∈X x
x
x ∈ X, then, as is well-known, a net (s i ) converges in such a product to s if and
only if s i (x) → s(x) in Y for each x, see Theorem A.5.2 (e). But, given that
Y is endowed with the discrete topology, this latter condition s i (x) → s(x)
holds if and only if s i (x) is eventually equal to s(x), as required.
•
Theorem 3.3.1 holds with X taken as B P (= B P,J ), where P is a normal
logic program, and Y taken as any set T of truth values, and in particular it
holds with T taken as T WO. With these choices, we obtain the following result, which is analogous to Proposition 3.2.10, but applies to normal programs
in general.
3.3.2 Proposition Let P be a normal logic program. Suppose that C is any
convergence class on I P,2 whose elements satisfy the condition stated in Theorem 3.3.1:
Topology and Logic Programming
lim i s i ≡ s (C) if and only if for each x ∈ X eventually s i (x) = s(x)
determines a convergence class on [X → Y ] whose associated topology Q is
the product of X copies of the discrete topology on Y .
Proof: We must verify that the conditions (1), (2), (3), and (4) in the definition of a convergence class, see Definition 3.1.2, hold with the given meaning
of lim i s i ≡ s (C).
(1) Suppose that s i = s for all i ∈ I is a constant net. Then s i (x) = s(x)
for all x and all i. Hence, for all x, eventually s i (x) = s(x), and so ((s i ), s) ∈ C.
(2) Suppose that ((s i ), s) ∈ C and that (t j ) j
is a subnet of (s i ) i . Let
∈J
∈I
x ∈ X be arbitrary, and let i 0 be such that s i (x) = s(x) for all i ≥ i 0 . Since
(t j ) is a subnet of (s i ), there is φ : J → I and j 0 ∈ J such that i 0 ≤ φ(j)
whenever j 0 ≤ j. But then, if j 0 ≤ j, we have t j (x) = s φ(j) (x) = s(x), and
hence ((t j ), s) ∈ C.
(3) Suppose that (s i ) i does not converge (C) to s. Then there is x ∈ X
∈I
and a cofinal subset J of I such that, whenever j ∈ J , we have s j (x) = s(x).
Let t j = s j for each j ∈ J . Then (t j ) is a subnet of (s i ), and clearly no subnet
of (t j ) converges (C) to s.
(4) Suppose that the conditions stated in (4) of Definition 3.1.2 all hold
and that lim m lim n x(m, n) ≡ s (C), where x : F
' → [X → Y ]. Consider the
net x ◦ r : F → [X → Y ]. Let y ∈ X be arbitrary. Since lim m lim n x(m, n) ≡
s (C), there is m 0 ∈ I such that, for all m ≥ m 0 , lim n x(m, n) ≡ s m (C)
for some s m ∈ [X → Y ], and lim m s m ≡ s (C). Therefore, for m ≥ m 0 ,
there is n m ∈ J m such that x(m, n)(
s y) = s m (y) for all n ≥ n m . But for
m ≥ m 0 , s m (y) = s(y). Define f ∈ m I J m by setting f (m) = n m ∈ J
∈
m
whenever m ≥ m 0 and otherwise letting f (m) ∈ J m be arbitrary. Suppose
(m, g) ≥ (m 0 , f ). Then m ≥ m 0 and g ≥ f so that g(m) ≥ f (m) = n m . But
then we have x(m, g(m))(y) = s m (y) = s(y). In other words, (x◦r)(m, g)(y) =
x(m, g(m))(y) = s(y) whenever (m, g) ≥ (m 0 , f ). Thus, (x◦r)(y) is eventually
equal to s(y), and hence x ◦ r converges (C) to s.
Finally, viewing [X → Y ] as the product
s
x
Y , where Y = Y for each
∈X x
x
x ∈ X, then, as is well-known, a net (s i ) converges in such a product to s if and
only if s i (x) → s(x) in Y for each x, see Theorem A.5.2 (e). But, given that
Y is endowed with the discrete topology, this latter condition s i (x) → s(x)
holds if and only if s i (x) is eventually equal to s(x), as required.
•
Theorem 3.3.1 holds with X taken as B P (= B P,J ), where P is a normal
logic program, and Y taken as any set T of truth values, and in particular it
holds with T taken as T WO. With these choices, we obtain the following result, which is analogous to Proposition 3.2.10, but applies to normal programs
in general.
3.3.2 Proposition Let P be a normal logic program. Suppose that C is any
convergence class on I P,2 whose elements satisfy the condition stated in Theorem 3.3.1: