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Mathematical Aspects of Logic Programming Semantics
(b) Whenever B 1 , B 2 ∈ B and x ∈ B 1 ∩ B 2 , there is B 3 ∈ B satisfying
x ∈ B 3 ⊆ B 1 ∩ B 2 .
Furthermore, any collection C of subsets of X is a subbase for some topology
on X, namely, the topology formed by taking all arbitrary unions of finite
intersections of elements of C.
A.2.8 Theorem Suppose that B is a collection of open sets in a topological
space X. Then B is a base for X if and only if, for each x ∈ X, the collection
B x = {B ∈ B | x ∈ B} is a neighbourhood base at x.
As noted in Definition A.2.1, the elements of τ are called the open sets in
the given topology on X. By definition, we call a subset F of X closed if its
complement, X \ F , is open. It follows immediately that ∅ and X are closed
sets, that any finite union of closed sets is itself closed, and that an arbitrary
intersection of closed sets is closed. Therefore, given an arbitrary subset E of
X, the intersection E of all the closed sets containing E is a closed set, the
smallest closed set containing E, and is called the closure of E. Clearly, a set
F is closed if and only if F = F . Dually, one defines the interior U
o of a
subset U of X to be the largest open set contained in U , and it is of course
the union of all the open sets contained in U . Moreover, it is also clear that a
set O is open if and only if O = O
o .
A closure operator (also known as a Kuratowski, or topological, closure
c
operator) on a set X is a mapping : P(X) → P(X), from the power set
P(X) of X into itself, subject to the following axioms.
(1) ∅
c = ∅.
(2) A ⊆ A
c for all A ⊆ X.
(3) (A ∪ B)
c = A
c ∪ B
c for all A, B ⊆ X.
(4) A
c = (A
c
)
c for all A ⊆ X.
Just as the notion of an open set can be taken as basic in defining topologies, so clearly can the notion of a closed set. More interesting is the fact that
closure can be taken as fundamental, and indeed the characteristic properties
of closure are precisely the four just stated in defining a closure operator, in
the following sense.
A.2.9 Theorem Let X be a non-empty set, and let
c : P(X) → P(X) be a
closure operator on X. Then τ = {X \ A | A ⊆ X, A = A
c
} is a topology on
X, called the topology associated with
c
, in which we have A = A
c for each
subset A of X. Thus, A
c is the topological closure in X of each subset A of
X with respect to the topology τ associated with
c .
Mathematical Aspects of Logic Programming Semantics
(b) Whenever B 1 , B 2 ∈ B and x ∈ B 1 ∩ B 2 , there is B 3 ∈ B satisfying
x ∈ B 3 ⊆ B 1 ∩ B 2 .
Furthermore, any collection C of subsets of X is a subbase for some topology
on X, namely, the topology formed by taking all arbitrary unions of finite
intersections of elements of C.
A.2.8 Theorem Suppose that B is a collection of open sets in a topological
space X. Then B is a base for X if and only if, for each x ∈ X, the collection
B x = {B ∈ B | x ∈ B} is a neighbourhood base at x.
As noted in Definition A.2.1, the elements of τ are called the open sets in
the given topology on X. By definition, we call a subset F of X closed if its
complement, X \ F , is open. It follows immediately that ∅ and X are closed
sets, that any finite union of closed sets is itself closed, and that an arbitrary
intersection of closed sets is closed. Therefore, given an arbitrary subset E of
X, the intersection E of all the closed sets containing E is a closed set, the
smallest closed set containing E, and is called the closure of E. Clearly, a set
F is closed if and only if F = F . Dually, one defines the interior U
o of a
subset U of X to be the largest open set contained in U , and it is of course
the union of all the open sets contained in U . Moreover, it is also clear that a
set O is open if and only if O = O
o .
A closure operator (also known as a Kuratowski, or topological, closure
c
operator) on a set X is a mapping : P(X) → P(X), from the power set
P(X) of X into itself, subject to the following axioms.
(1) ∅
c = ∅.
(2) A ⊆ A
c for all A ⊆ X.
(3) (A ∪ B)
c = A
c ∪ B
c for all A, B ⊆ X.
(4) A
c = (A
c
)
c for all A ⊆ X.
Just as the notion of an open set can be taken as basic in defining topologies, so clearly can the notion of a closed set. More interesting is the fact that
closure can be taken as fundamental, and indeed the characteristic properties
of closure are precisely the four just stated in defining a closure operator, in
the following sense.
A.2.9 Theorem Let X be a non-empty set, and let
c : P(X) → P(X) be a
closure operator on X. Then τ = {X \ A | A ⊆ X, A = A
c
} is a topology on
X, called the topology associated with
c
, in which we have A = A
c for each
subset A of X. Thus, A
c is the topological closure in X of each subset A of
X with respect to the topology τ associated with
c .