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60
26. December 8th: Topology IV.
26.1
.
Prove that
R
with the Zariski topology
is connected.
Answer:
Let
S
⊆
R
be any set other than
∅
and
R
.T
h
e
n
S
and
R°
S
cannot
both be fnite, since otherwise
R
itselF would be fnite. It Follows that
S
cannot both
be closed and open in the Zariski topology. So there is no set as required by the
defnition oF
disconnected.
It Follows that
R
is connected.
°
26.2
.
Let
f
:
T
1
→
T
2
be a surjective continuous Function between topological
spaces. Prove that iF
T
1
is connected, then
T
2
is connected.
Answer:
We can prove the equivalent (contrapositive) statement: iF
T
2
is discon
nected, then
T
1
must also be disconnected. ±or this, let
S
⊆
T
2
be a set that is
both closed and open, and such that
S
°
=
∅
,
S
°
=
T
2
.S
in
c
e
f
is surjective, it Follows
that
f
−
1
(
S
)
°
=
∅
,
f
−
1
(
S
)
°
=
T
1
. Also,
f
−
1
(
T
2
°
S
)=
T
1
°
f
−
1
(
S
). (Make sure you
are able to veriFy all these claims!) Since
S
is open and
f
is continuous,
f
−
1
(
S
)is
open. Since
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 Fall '11
 Aluffi

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