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Serge Ballif
MATH 527 Homework 1
September 7, 2007
Problem 1.
Let
X
be a topological space, and
A
be its subset. Is it true that
A
0
, the set of limit
points of
A
is always closed in
X
?
It is not true in general. For a counterexample, we can take
X
=
{
a,b
}
with
the indiscrete topology
T
=
{
X,
∅}
. A point
x
in
X
is a limit point of a subset
A
iﬀ every open set containing
x
also contains a point of
A
other than
x
itself.
Consider the subset
A
=
{
a
}
. The point
b
is a limit point of
A
, because every
open set containing
b
has
A
as a subset. However,
a
is not a limit point of
A
, since
no open set containing
a
contains any point of
A
other than
a
. Thus
A
0
=
{
b
}
,
which is not closed in
X
.
Problem 2.
Construct a map
f
:
R
→
R
that is continuous at exactly one point.
Let
f
:
R
→
R
be given by
f
(
x
) =
(
x
if
x
is rational

x
if
x
is irrational
We claim that
f
is continuous only at the point
x
= 0. A function
f
:
R
→
R
is
continuous at a point
x
iﬀ for every sequence (
x
i
)
∞
i
=1
that converges to
x
we also
have the sequence (
f
(
x
i
))
∞
i
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 Fall '07
 ROTMAN,REGINA
 Math, Logic, Topology

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