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Miklós Schweitzer
1982 Miklós Schweitzer
1
1
Part of
1982 Miklós Schweitzer
Problems
(1)
Miklos Schweitzer 1982_1
Source:
1/31/2009
A map
F
:
P
(
X
)
→
P
(
X
)
F : P(X) \rightarrow P(X)
F
:
P
(
X
)
→
P
(
X
)
, where
P
(
X
)
P(X)
P
(
X
)
denotes the set of all subsets of
X
X
X
, is called a
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
c
l
o
s
u
r
e
o
p
e
r
a
t
i
o
n
<
/
s
p
a
n
>
<span class='latex-italic'>closure operation</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
c
l
os
u
reo
p
er
a
t
i
o
n
<
/
s
p
an
>
on
X
X
X
if for arbitrary
A
,
B
⊂
X
A,B \subset X
A
,
B
⊂
X
, the following conditions hold: (i)
A
⊂
F
(
A
)
;
A \subset F(A);
A
⊂
F
(
A
)
;
(ii)
A
⊂
B
⇒
F
(
A
)
⊂
F
(
B
)
;
A \subset B \Rightarrow F(A) \subset F(B);
A
⊂
B
⇒
F
(
A
)
⊂
F
(
B
)
;
(iii) F(F(A))\equal{}F(A). The cardinal number \min \{ |A| : \;A \subset X\ ,\;F(A)\equal{}X\ \} is called the
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
d
e
n
s
i
t
y
<
/
s
p
a
n
>
<span class='latex-italic'>density</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
d
e
n
s
i
t
y
<
/
s
p
an
>
of
F
F
F
and is denoted by
d
(
F
)
d(F)
d
(
F
)
. A set
H
⊂
X
H \subset X
H
⊂
X
is called
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
d
i
s
c
r
e
t
e
<
/
s
p
a
n
>
<span class='latex-italic'>discrete</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
d
i
scre
t
e
<
/
s
p
an
>
with respect to
F
F
F
if u \not \in F(H\minus{}\{ u \}) holds for all
u
∈
H
u \in H
u
∈
H
. Prove that if the density of the closure operation
F
F
F
is a singular cardinal number, then for any nonnegative integer
n
n
n
, there exists a set of size
n
n
n
that is discrete with respect to
F
F
F
. Show that the statement is not true when the existence of an infinite discrete subset is required, even if
F
F
F
is the closure operation of a topological space satisfying the
T
1
T_1
T
1
separation axiom. A. Hajnal
topology