Solution Sketches and Comments, Math 74 HW4
“The set is open”
is not equivalent to
“The set is not closed,”
and similarly, “The
set is closed” is not equivalent to “The set is not open.” You actually showed this in Problem
4.1 by proving that a set can be both closed and open. The terminology might be confusing
and even more so when you’re thinking in terms of
R
, since typical sets like (
a, b
) and [
a, b
]
are either open and not closed or closed and not open.
A lot of people argued in Problem 4.10 that, because their example of a set is open,
it must be not closed, which isn’t enough.
In Math 104 (or 141 or 142), you’ll learn that
the property of
R
being
connected
implies that the only subsets of
R
which are both open
and closed are
R
itself and
∅
– so for a correct justiﬁcation, you needed to mention that
(
a, b
) is a
proper nonempty
open subset of
R
, implying (
a, b
) cannot be closed. (From what’s
been covered in Math 74, you can show that
a /
∈
(
a, b
) is a limit point, so (
a, b
) is not closed.)
Corrections to HW3 Solutions:
The function
h
given in Problem 3.2 isn’t bijective
because nothing is mapped to 0. The correct function should be
h
(
x
) =
±
f
(
x
)
,
x
∈
A,

g
(
x
) + 1
,
x
∈
B.
Also, the deﬁnition of continuity in Problem 3.9 is missing “
< ε
” at the end of the line.
4.1
Proposition.
Let
d
be a metric on
X
. For any
x
∈
X
,
{
x
}
is a closed set, and
X,
∅
are
clopen
sets – both closed and open.
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