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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.