56
24. December 1st: Topology II.
24.1
.
I have allowed the endpoints of the open intervals considered in Defini
tion 24.5 to be
±
∞
. Prove that it is not necessary to do this, in the sense that one
defines the same topology by ‘just’ taking unions of bounded open intervals (
a, b
)
with
a, b
∈
R
(and
a < b
).
Answer:
The point here is simply that the extended open intervals (
∞
, b
),
(
a,
∞
), and (
∞
,
∞
) are themselves unions of bounded open intervals:
(
a,
∞
) =
∪
b>a
(
a, b
)
(
∞
, b
) =
∪
a<b
(
a, b
)
(
∞
,
∞
) =
∪
a>
0
(
a, a
)
.
Thus every ‘union of open intervals’ as in Definition 24.5 can in fact be rewritten as
a union of bounded open intervals.
24.2
.
Let
R
stan
, resp.
R
Zar
be the topological spaces determined by the standard,
resp. Zariski topology on
R
. Prove that the identity
R
→
R
is a continuous function
R
stan
→
R
Zar
, and that it is not a homeomorphism.
Answer:
We have to verify that if a set is open in the Zariski topology, then it is
open in the standard topology; and that the converse is not necessarily true.
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 Fall '11
 Aluffi
 Topology, Empty set, Metric space, Open set, Topological space, Closed set

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