79
Topology and Logic Programming
(a) A net (I i ) of interpretations converges to an interpretation I if and only
if, for each ground atom A, we have that I i (A) is eventually equal to I(A).
(b) I(X, T ) is a totally disconnected Hausdorff space.
(c) I(X, T ) is compact if and only if T is a finite set.
(d) I(X, T ) is metrizable
13 if and only if D is countable.
(e) I(X, T ) is second countable if and only if D and T are both countable.
(f) Suppose that D is denumerable and that T is finite. Then I(X, T ) is
homeomorphic to the Cantor set in the closed unit interval within the
real line.
Proof: Statement (a) follows immediately from Theorem 3.3.1, and all the
remaining statements follow from general and well-known results concerning product spaces, see the Appendix. Specifically, they can be found in
[Willard, 1970], where unfamiliar terms are also defined, as follows: for (b),
see Page 72, Theorem 13.8, and Page 210, Theorem 29.3; (c) follows from Tychonoff’s theorem (Page 120, Theorem 17.8) and the fact that a discrete space
is compact if and only if it is finite; for (d), see Page 161, Theorem 22.3; for
(e), see Page 108, Theorem 16.2; and finally, for (f), see Page 217, Corollary
30.6.
•
Because of Part (f) of Theorem 3.3.4, we refer to the topology Q as the
Cantor topology.
Notice that I(X, T ) is a Hausdorff space in the topology Q, and hence
the limit of any net convergent in Q is unique, see Theorem A.4.2, unlike the
situation in the Scott topology where a convergent net has many limits in
general, as shown by Example 3.2.7.
In the case of two-valued interpretations, we have the following result. It
follows immediately from Part (a) of Theorem 3.3.4 and will be used quite
often later on.
3.3.5 Proposition A net (I i ) of interpretations in I P,2 converges to I in the
topology Q if and only if whenever A ∈ I, eventually A ∈ I i , and whenever
A ∈ I, eventually A ∈ I i . Moreover, the unique limit I coincides with the set
{A ∈ B P | A eventually belongs to I i }.
The following example illustrates Proposition 3.3.5.
13 Metrics are defined in Section 4.2 and studied extensively in Chapter 4. A topological
space is said to be metrizable if its open sets can be defined in terms of some metric as
discussed in Section 4.1. The representation of Q as a product topology makes it easy to
determine metrics for Q, see [Seda, 1995].
Topology and Logic Programming
(a) A net (I i ) of interpretations converges to an interpretation I if and only
if, for each ground atom A, we have that I i (A) is eventually equal to I(A).
(b) I(X, T ) is a totally disconnected Hausdorff space.
(c) I(X, T ) is compact if and only if T is a finite set.
(d) I(X, T ) is metrizable
13 if and only if D is countable.
(e) I(X, T ) is second countable if and only if D and T are both countable.
(f) Suppose that D is denumerable and that T is finite. Then I(X, T ) is
homeomorphic to the Cantor set in the closed unit interval within the
real line.
Proof: Statement (a) follows immediately from Theorem 3.3.1, and all the
remaining statements follow from general and well-known results concerning product spaces, see the Appendix. Specifically, they can be found in
[Willard, 1970], where unfamiliar terms are also defined, as follows: for (b),
see Page 72, Theorem 13.8, and Page 210, Theorem 29.3; (c) follows from Tychonoff’s theorem (Page 120, Theorem 17.8) and the fact that a discrete space
is compact if and only if it is finite; for (d), see Page 161, Theorem 22.3; for
(e), see Page 108, Theorem 16.2; and finally, for (f), see Page 217, Corollary
30.6.
•
Because of Part (f) of Theorem 3.3.4, we refer to the topology Q as the
Cantor topology.
Notice that I(X, T ) is a Hausdorff space in the topology Q, and hence
the limit of any net convergent in Q is unique, see Theorem A.4.2, unlike the
situation in the Scott topology where a convergent net has many limits in
general, as shown by Example 3.2.7.
In the case of two-valued interpretations, we have the following result. It
follows immediately from Part (a) of Theorem 3.3.4 and will be used quite
often later on.
3.3.5 Proposition A net (I i ) of interpretations in I P,2 converges to I in the
topology Q if and only if whenever A ∈ I, eventually A ∈ I i , and whenever
A ∈ I, eventually A ∈ I i . Moreover, the unique limit I coincides with the set
{A ∈ B P | A eventually belongs to I i }.
The following example illustrates Proposition 3.3.5.
13 Metrics are defined in Section 4.2 and studied extensively in Chapter 4. A topological
space is said to be metrizable if its open sets can be defined in terms of some metric as
discussed in Section 4.1. The representation of Q as a product topology makes it easy to
determine metrics for Q, see [Seda, 1995].