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The following is a list of deﬁnitions and theorem from class. To use this
list eﬀectively, don’t just read the list. See if you can state the deﬁnitions
without looking. Rewrite the proofs yourself, making sure that you follow
the logic. There may be some typos. See if you can ﬁnd any.
Deﬁnition.
A
is
open
if
∀
x
∈
A,
∃
± >
0
,V
±
(
x
)
⊂
A.
Deﬁnition.
x
∈
R
is a
limit point
of
A
if
∀
± >
0
,V
±
(
x
)
∩
A
\ {
x
} 6
=
∅
.
The set of limit points is denoted
L
A
.
Deﬁnition.
x
∈
A
is an
isolated point
of
A
if
∃
± >
0
,V
±
(
x
)
∩
A
=
{
x
}
.
Deﬁnition.
A
is
closed
if
L
A
⊂
A
.
Theorem 1
Any point in
A
that is not an isolated point of
A
is a limit point
of
A
.
Proof.
Let
x
∈
A
.
x
is not an isolated point of
A
=
⇒ ∀
± >
0
,
¬
[
{
x
}
=
V
±
(
x
)
∩
A
]
=
⇒ ∀
± >
0
,V
±
(
x
)
∩
A
\ {
x
} 6
=
∅
contains an element other than
x
.
Theorem 2
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This note was uploaded on 11/27/2011 for the course MCH 108 taught by Professor Penelopekirby during the Fall '08 term at FSU.
 Fall '08
 PenelopeKirby

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