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3.
φ
ψ
=
Int(
R\
φ
ψ
) ;
4.
) ,
where Int(S) is the interior of the set S (i.e., the largest open subset of S ). The set
S is open if Int(S)
¬
φ
=
Int(
R\
φ
S (the operation Int
corresponds to the universal modal operator denoted usually by '
=
S , and closed if Clo(S)
R\
Int(
R\
S)
=
', while Clo to
its dual existential modal operator '
').
In fact, from the algebraic point of view, we obtain a Boolean algebra
P
, 0 , 1
(
R
),
,
,
with 0 equal to the empty set and 1 equal to
R
, and
the set-complement
(
S R\ S for any subset S P ( R ) ), so that ( P ( R ), , , , Int, 0 , 1 ) is a topo-
Boolean (or S4) algebra (which satisfies normal modal properties for the multiplica-
tive modal operator, Int(S S ) =
Int(S ), Int( 1 ) =
Int(S)
1, and “open” properties
Int(Int(S)) , for any S,S P ( R ) ), that is, an N -modal algebra
Int(S) S , Int(S)
for N =
1.
Hence, for
φ =−{
0
}=R\{
0
}
we obtain
¬ φ =
Int( {
0
} ) =∅
, thus the ex-
cluded middle does not hold,
φ ∨¬ φ = ( −{
0
} ) ∪∅=R
. Consequently, this infi-
nite Heyting algebra ( O ( R ), , , , Int( R\
_
_ ), Int( R\
_ ), , R ) is complete for
IPC , such that [ 21 ], i I S i = i I S i and i I S i =
Int( i I S i ) .
Notice that for the dual monotonic “closure” operator Clo we have the dual prop-
erties:
1. Clo(S S ) =
Clo(S ) , Clo( 0 ) =
Clo(S)
0; (modal additivity property)
2. S
Clo(S) , Clo(Clo(S))
Clo(S) ; (closure property)
3. i I S i =
Clo( i I S i ) and i I S i = i I S i .
Let us consider the topological interpretation of WMTL:
R
O
R
Example 46
(C LOSURE OPERATION IN THE TOPOLOGICAL SPACE (
,
(
)) )
R
O W (
R
O W (
R
Let us consider the topological space (
,
)) where
) is the set of all
={
∈R|
}
O W (
R
={
⊆R|
}∪{∅}⊂
open intervals (a,b)
x
a<x<b
,
)
(a,b)
a<b
O
R
:
O W (
R
(
) . So, the valuation
_
PR
) is extended inductively for all formu-
lae by:
1.
=
φ
ψ
φ
ψ
;
2.
Clo
φ
ψ
=
φ
ψ
= min glb
, glb
, max lub
, lub
;
φ
ψ
φ
ψ
3.
φ
ψ
=
Clo(Int(
R\
φ
ψ
)) ;
4.
)) ,
where Int(S) is the interior of the set S (i.e., the largest open subset of S ), and
Clo(S) is the least open set (open interval) in
¬
φ
=
Clo(Int(
R\
φ
O W (
R
) such that S
Clo(S) and for
each S
O W (
R
), Clo(S)
=
S (so that Clo(
)
=∅
and Clo(
R
)
=R
). It is easy to
verify that Clo is a closure topological operator, and that
¬
φ
=R
if
φ
=R
;
otherwise.
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