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=
n
≥
0
R
n
(
1. If R is FP, then lfp
(
R
)
∅
)
;
=
n
≥
0
R
n
(
X
)
.
2. If R is FC, then gfp
(
R
)
Proof
Combine Propositions
2.1
and
2.2
.
Intuitively, the set
R
0
(
; the set
R
1
(
=
R
(
∅
=∅
∅
∅
) consisting of all the conclu-
sions of instances of axioms. In general, the set
R
n
+
1
(
)
)
∅
) contains all objects which
immediately follow by ground rules with premises in
R
n
(
). The above corollary
states that if
R
is FP, each element in the set
lfp
(
R
) can be derived via a derivation
tree of finite depth whose leaves are instances of axioms; if
R
is FC, elements in
gfp
(
R
) can always be destructed as conclusions of ground rules whose premises can
also be destructed similarly.
∅
2.3
Topological Spaces
In this section, we review some fundamental concepts in general topology such as
continuous functions, compact sets and some other related properties. They will be
used in Chap. 6.
Definition 2.6
Let
X
be a nonempty set. A collection
T
of subsets of
X
is a
topology
on
X
iff
T
satisfies the following axioms.
1.
X
and
;
2. The union of any number of sets in
∅
belong to
T
T
belongs to
T
;
3. The intersection of any two sets in
T
belongs to
T
.
T
T
The members of
are called
open sets
, and the pair (
X
,
) is called a
topological
space
.
Example 2.7
The collection of all open intervals in the real line
R
forms a topology,
which is called the
usual topology
on
R
.
) be a topological space. A point
x
∈
X
is an
accumulation point
or
limit point
of a subset
Y
of
X
iff every open set
Z
containing
x
must also contain a
point of
Y
different from
x
, that is,
Let (
X
,
T
Z
open,
x
∈
Z
implies
(
Z
\{
x
}
)
∩
Y
=∅
.
Definition 2.7
A subset
Y
of a topological space (
X
,
T
)is
closed
iff
Y
contains
each of its limit points.
Definition 2.8
). The
closure
of
Y
is
the set of all limit points of
Y
. We say
Y
is
dense
if the closure of
Y
is
X
, i.e. every
point of
X
is a limit point of
Y
.
It is immediate that
Y
coincides with its closure if and only if
Y
is closed.
Definition 2.9
Let
Y
be a subset of a topological space (
X
,
T
) is said to be
separable
if it contains a
countable dense subset; that is, if there exists a finite or denumerable subset
Y
of
X
such that the closure of
Y
is the entire space.
A topological space (
X
,
T
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