Prof. D. P.Patil, Department of Mathematics, Indian Institute of Science, Bangalore
AugustDecember 2004
MA231 Topology
9. Convergence — Inadquacy of sequences, Nets
1
)
——————————————————————
October 11, 2004
Eliakim Hastings Moore
†
(18621932)
9.1. a).
(Topology of firstcountable spaces)Let
X
and
Y
be firstcountable topological
spaces. Then
(1) A subset
U
⊆
X
is open if and only if whenever a sequence
(x
n
)
in
X
converges to a point
x
∈
U
,
then there exists
n
0
∈
N
such that
x
n
∈
U
for all
n
≥
n
0
.
(2) A subset
F
⊆
X
is closed if and only if whenever a sequence
(x
n
)
is contained in
F
and
x
n
converges
to a point
x
∈
X
, then
x
∈
F
.
(3) A map
f
:
X
→
Y
is continuous if and only if whenever a sequence
(x
n
)
converges to
x
in
X
, then
the image sequence
f(x
n
)
converges to
f(x)
in
Y
.
b).
Which of the above properties ain the above part a) hold for an uncountable set
X
with the cofinite
topology ?
9.2.
a).
In
R
R
, let
E
={
f
∈
R
R

f(x)
∈{
0
,
1
}
and
f(x)
=
0 only finitely often
}
and let
g
∈
R
R
be the constant function 0 . Then in the product topology on
R
R
,
g
∈
E
. Construct a net
(f
λ
)
in
E
which converges to
g
.
b).
Let
(X, d)
be a mtric space and let
x
0
∈
X
. Define
≤
on
X
\{
x
0
}
by
y
≤
z
if and only if
d(y,x
0
)
≥
d(z,x
0
)
. With this relation
X
\{
x
0
}
is a directed set and hence any map
f
:
X
→
Y
into
another metric space
Y
defines (by restricting
f
to
X
\{
x
0
}
) a net in
Y
. Show that this net converges
to a point
y
0
∈
Y
if and only if lim
x
→
x
0
f(x)
=
y
0
.
9.3.
(Cluster points) Let
X
be a topological space and let
x
∈
X
. We say that
x
isa cluster
point ofanet
(x
λ
)
λ
∈
3
in
X
if for each nhood
U
of
x
and for each
λ
0
∈
3
, there exists
λ
∈
3
with
λ
≥
λ
0
such that
x
λ
∈
U
. Let
(x
λ
)
λ
∈
3
be a net in a topological space
X
.
a).
Show that
(x
λ
)
λ
∈
3
has a cluster point
x
∈
X
if and only if
(x
λ
)
λ
∈
3
has a subnet which converges to
x
.(
Proof :
Suppose that
x
is a cluster point of the net
(x
λ
)
. Let
M
:
={
(λ, U )

λ
∈
3, U
∈
U
x
with
x
λ
∈
U
}
.
The set
M
is directed by the relation :
(λ
1
,U
1
)
≤
(λ
2
,U
2
)
if and only if
λ
1
≤
λ
2
and
U
2
⊆
U
1
. The map
ϕ
:
M
→
3
defined by
ϕ((λ, U ))
=
λ
is increasing and cofinal in
3
and hence defines a subnet of
(x
λ
)
which
converges to
x
. Conversely, suppose that
ϕ
:
M
→
3
defines a subset of
(x
λ
)
which converges to
x
. Then for
each nhood
U
of
x
, there is some
µ
U
∈
M
such that
µ
≥
µ
U
implies that
x
ϕ(µ)
∈
U
. Now, suppose that a nhood
U
of
x
and a point
λ
0
∈
3
are given. Since
ϕ(M)
is cofinal in
3
, there is some
µ
0
∈
M
such that
ϕ(µ
0
)
≥
λ
0
. But
there is also
µ
U
∈
M
such that
µ
≥
µ
U
implies that
x
ϕ(µ)
∈
U
. Choose
µ
∗
∈
M
such that
µ
∗
≥
µ
0
and
µ
∗
≥
µ
U
.
Then