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Mathematical Aspects of Logic Programming Semantics
The sense, mentioned earlier, in which nets can describe all basic topological notions can now be clarified.
A.3.5 Theorem Let X and Y be topological spaces. Then the following
statements hold.
(a) Let E ⊆ X. Then x ∈ E if and only if there is a net (s i ) in E such that
s i → x.
(b) A subset O of X is open if and only if, whenever x ∈ O and (s i ) is a net
such that s i → x, we have that (s i ) is eventually in O.
(c) A subset F of X is closed if and only if, whenever (s i ) is a net in F and
s i → x, we have x ∈ F .
(d) A function f : X → Y is continuous at x ∈ X if and only if, whenever
s i → x in X, we have f (s i ) → f (x) in Y .
Proof: We include a proof of (b) here since we have specific need of the result.
Suppose that O is open, that x ∈ O, and that s i → x. Then it is clear from
the definition of net convergence that (s i ) is eventually in O.
Conversely, assuming the stated condition, we show that O contains a
neighbourhood of each of its points and hence is open. Let x ∈ O, and let
U x be the neighbourhood system of x. Let I = {(y, U ) | y ∈ U ∈ U x } ordered by (y 1 , U 1 ) ≤ (y 2 , U 2 ) if and only if U 2 ⊆ U 1 . Then it is easy to see
that the ordering ≤ directs I and also that the net s : I → X defined by
s(y, U ) = y converges to x. By our current hypothesis, this net is eventually
in O. Let (y 0 , U 0 ) be such that s (y,U ) = y ∈ O whenever (y 0 , U 0 ) ≤ (y, U ).
Since (y 0 , U 0 ) ≤ (y, U 0 ) for all y ∈ U 0 , we conclude that x ∈ U 0 ⊆ O, as
required.
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A.4 Separation Properties and Compactness
It is important to have sufficiently many open sets to be able to distinguish,
in some way, between points in a topological space by means of the open sets.
This is usually done by means of the following axioms.
A.4.1 Definition Let X be a topological space.
(1) We call X a T 0 -space if, whenever x and y are distinct points of X, there
is an open set containing one but not the other.
(2) We call X a T 1 -space if, whenever x and y are distinct points of X, there
is a neighbourhood of each not containing the other.
Mathematical Aspects of Logic Programming Semantics
The sense, mentioned earlier, in which nets can describe all basic topological notions can now be clarified.
A.3.5 Theorem Let X and Y be topological spaces. Then the following
statements hold.
(a) Let E ⊆ X. Then x ∈ E if and only if there is a net (s i ) in E such that
s i → x.
(b) A subset O of X is open if and only if, whenever x ∈ O and (s i ) is a net
such that s i → x, we have that (s i ) is eventually in O.
(c) A subset F of X is closed if and only if, whenever (s i ) is a net in F and
s i → x, we have x ∈ F .
(d) A function f : X → Y is continuous at x ∈ X if and only if, whenever
s i → x in X, we have f (s i ) → f (x) in Y .
Proof: We include a proof of (b) here since we have specific need of the result.
Suppose that O is open, that x ∈ O, and that s i → x. Then it is clear from
the definition of net convergence that (s i ) is eventually in O.
Conversely, assuming the stated condition, we show that O contains a
neighbourhood of each of its points and hence is open. Let x ∈ O, and let
U x be the neighbourhood system of x. Let I = {(y, U ) | y ∈ U ∈ U x } ordered by (y 1 , U 1 ) ≤ (y 2 , U 2 ) if and only if U 2 ⊆ U 1 . Then it is easy to see
that the ordering ≤ directs I and also that the net s : I → X defined by
s(y, U ) = y converges to x. By our current hypothesis, this net is eventually
in O. Let (y 0 , U 0 ) be such that s (y,U ) = y ∈ O whenever (y 0 , U 0 ) ≤ (y, U ).
Since (y 0 , U 0 ) ≤ (y, U 0 ) for all y ∈ U 0 , we conclude that x ∈ U 0 ⊆ O, as
required.
•
A.4 Separation Properties and Compactness
It is important to have sufficiently many open sets to be able to distinguish,
in some way, between points in a topological space by means of the open sets.
This is usually done by means of the following axioms.
A.4.1 Definition Let X be a topological space.
(1) We call X a T 0 -space if, whenever x and y are distinct points of X, there
is an open set containing one but not the other.
(2) We call X a T 1 -space if, whenever x and y are distinct points of X, there
is a neighbourhood of each not containing the other.