AN INTRODUCTION TO TOPOLOGY
Fall Semester 2007
0.
Set Theory
In set theory, there are two undefined or primitive concepts:
set
and
is an element of
.
Taken together, they are the first principals of set theory. Hence their meaning does not
derive from other more fundamental concepts, but from their relationship to each other: A
set
is a collection of objects, each of which
is an element of
the set.
We write
x
∈
S
to
denote that
x
is an element of
S
.
In keeping with common English usage, the word “set” is roughly synonymous with
“family,” “collection,” “aggregate,” and “class,” and the phrase “is and element of” is used
interchangeably with “is a member of,” “is contained by,” and “belongs to.” Often the
members of a set are called “points.” However, we will have many occasions to deal with
collections whose members are themselves sets. In this case, it seems inappropriate, though
not technically incorrect, to call these sets “points”.
Definition.
A set
A
is a
subset
of a set
B
, denoted
A
⊂
B
if and only if every element of
A
is also an element of
B
;
A
is a
proper subset
of
B
if and only
A
is a subset of
B
, but
B
is not a subset of
A
. Two sets
A
and
B
are
equal
if and only if
A
is a subset of
B
and
B
is
a subset of
A
.
Suppose
A
is a collection of sets.
The
union
of
A
, denoted
A
*
or
∪A
, is the set to
which a point belongs if and only if it belongs to some member of the collection
A
. The
intersection
of
A
, denoted
∩A
, is the set to which a point belongs if and only if it belongs
to each member of the collection
A
.
If
A
=
{
A, B
}
, then
A
*
and
∩A
are conventionally written
A
∪
B
and
A
∩
B
respectively.
Similar conventions exist when
A
is any finite or countably infinite collection.
Notation.
Sometimes the members of a collection of sets are indexed by another set. If
Λ is a set and, for each
λ
∈
Λ,
A
λ
is a set, then the collection
{
A
λ
:
λ
∈
Λ
}
is said to be
indexed
by the set Λ. In this case,
{
A
λ
:
λ
∈
Λ
}
*
may also be denoted by
∪
λ
∈
Λ
A
λ
or just
∪
A
λ
. Similarly,
∩{
A
λ
:
λ
∈
Λ
}
may also be denoted
∩
λ
∈
Λ
A
λ
or
∩
A
λ
.
Sometimes it is convenient to index the members of a collection.
For example, it is
natural to index the collection of all sets of the form
{
1
,
2
, ..., n
}
with the natural numbers
because of the clear connection between the members of the collection and the members of
the indexing set
N
. Since the set of rational numbers are countable, it is sometimes useful to
index them with
N
and write
Q
=
{
r
1
, r
2
, r
3
, ...
}
. Other times, indexing a set or a collection
merely complicates things by introducing superfluous notation. Consequently, one should
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not index the members of a set or a collection without good reason. Using
{
A
i
}
to denote
a collection that may not be countable is unforgivably insidious.
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 Spring '08
 Ryden
 Set Theory, Topology, Topological space, xa, General topology, Hausdorff, A. Theorem

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