240
Mathematical Aspects of Logic Programming Semantics
topology. Indeed, the sets π
−1 (U i ) form a subbase for this topology, where
i
i ∈ I and U i is an open set in X i . It is immediate that each of the projections
π i is continuous relative to the product topology and the given topology on
the factor X i .
s
Subspaces of X and products
X i inherit certain properties enjoyed
i∈I
by X and the X i , respectively, as one would expect. We summarize next the
ones relevant to our needs in the following theorem.
A.5.2 Theorem The following statements hold.
(a) Subspaces of T 0 or Hausdorff spaces are T 0 or Hausdorff, respectively.
(b) If X is compact and S is a closed subset of X, then S is compact (as a
topological space in its own right). If X is Hausdorff and S is compact,
then S is a closed subset of X.
(c) A non-empty product
s
i
X i is T 0 or Hausdorff if and only if each factor
∈I
space X i is T 0 or Hausdorff, respectively.
(d) (Tychonoff ’s theorem) A non-empty product
and only if each factor space is compact.
s
i∈I X i is compact if
(e) A net (f λ ) in a product space
s
i
X i converges to f if and only if, for
∈I
each index i ∈ I, we have π i (f λ ) → π i (f ) in X i .
A.6 The Scott Topology
We present here the proofs of those results which were simply stated in
Chapter 3 concerning the Scott topology. In fact, our development constitutes
a treatment of the Scott topology from the point of view of convergence.
Unless stated to the contrary, (D, [) will denote throughout some fixed, but
arbitrary, domain with set D c of compact elements.
A.6.1 Proposition Suppose that A ⊆ D is a directed set. Then A is a net in
D, and, as a net, we have that A →
A in the Scott topology. In particular,
for each s ∈ D, approx(s) → s in the Scott topology.
Proof: Write A = {a i | i ∈ I} for some index set I, which we identify with A.
Then I is clearly directed by the ordering ≤ obtained by restricting [ to A.
Therefore, the inclusion map I → D is a net in D. Let A = A and suppose
that O is a neighbourhood of A in the Scott topology. Thus, A ∈ O, and
hence there exists some index i 0 such that a i0 ∈ O. But O is
upwards closed,
and therefore a i ∈ O whenever i 0
≤ i. Thus, A → A, as required.
•
Mathematical Aspects of Logic Programming Semantics
topology. Indeed, the sets π
−1 (U i ) form a subbase for this topology, where
i
i ∈ I and U i is an open set in X i . It is immediate that each of the projections
π i is continuous relative to the product topology and the given topology on
the factor X i .
s
Subspaces of X and products
X i inherit certain properties enjoyed
i∈I
by X and the X i , respectively, as one would expect. We summarize next the
ones relevant to our needs in the following theorem.
A.5.2 Theorem The following statements hold.
(a) Subspaces of T 0 or Hausdorff spaces are T 0 or Hausdorff, respectively.
(b) If X is compact and S is a closed subset of X, then S is compact (as a
topological space in its own right). If X is Hausdorff and S is compact,
then S is a closed subset of X.
(c) A non-empty product
s
i
X i is T 0 or Hausdorff if and only if each factor
∈I
space X i is T 0 or Hausdorff, respectively.
(d) (Tychonoff ’s theorem) A non-empty product
and only if each factor space is compact.
s
i∈I X i is compact if
(e) A net (f λ ) in a product space
s
i
X i converges to f if and only if, for
∈I
each index i ∈ I, we have π i (f λ ) → π i (f ) in X i .
A.6 The Scott Topology
We present here the proofs of those results which were simply stated in
Chapter 3 concerning the Scott topology. In fact, our development constitutes
a treatment of the Scott topology from the point of view of convergence.
Unless stated to the contrary, (D, [) will denote throughout some fixed, but
arbitrary, domain with set D c of compact elements.
A.6.1 Proposition Suppose that A ⊆ D is a directed set. Then A is a net in
D, and, as a net, we have that A →
A in the Scott topology. In particular,
for each s ∈ D, approx(s) → s in the Scott topology.
Proof: Write A = {a i | i ∈ I} for some index set I, which we identify with A.
Then I is clearly directed by the ordering ≤ obtained by restricting [ to A.
Therefore, the inclusion map I → D is a net in D. Let A = A and suppose
that O is a neighbourhood of A in the Scott topology. Thus, A ∈ O, and
hence there exists some index i 0 such that a i0 ∈ O. But O is
upwards closed,
and therefore a i ∈ O whenever i 0
≤ i. Thus, A → A, as required.
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