Basic Topology
Finite, Countable, and Uncountable Sets
2.1 Definition
If
A
and
B
are sets, a
function
f
from
A
into
B
is a subset of
A
×
B
such that for every
a
in
A
there exists a unique
b
in
B
such that (
a, b
)
∈
f
. The uniqueness condition can be formulated explicitly
as follows: if (
a, b
)
∈
f
and (
a, c
)
∈
f
, then
b
=
c
.
We call
A
the
domain
, and
B
the
codomain
, of
f
.
The function
f
from
A
to
B
is usually denoted
f
:
A
→
B
, and for each
a
∈
A
, the unique
b
∈
B
such that
(
a, b
)
∈
f
is denoted by
f
(
a
).
2.2 Definition
Let
A
and
B
be two sets and let
f
be a mapping of
A
into
B
. If
E
⊆
A
, the
image
of
E
under
f
is defined as
f
(
E
) =
{
f
(
x
):
x
∈
E
}
. The
range
of
f
is
f
(
A
). If
f
(
A
) =
B
, we say that
f
maps
onto
B
. If
C
⊆
B
, the
preimage
of
C
under
f
is defined as
f
−
1
(
C
) =
{
x
:
f
(
x
)
∈
C
}
. We say that
f
is
onetoone
if
x
1
negationslash
=
x
2
implies that
f
(
x
1
)
negationslash
=
f
(
x
2
) for all
x
1
, x
2
∈
A
.
2.3 Definition
If there exists a onetoone mapping of
A
onto
B
, we say that
A
and
B
can be put in
onetoone correspondence
, or that
A
and
B
have the same
cardinal number
, or briefly, that
A
and
B
are
equivalent
, and we write
A
∼
B
. Onetoone correspondence is an equivalence relation:
It is reflexive:
A
∼
A
.
It is symmetric: If
A
∼
B
, then
B
∼
A
.
It is transitive: If
A
∼
B
and
B
∼
C
, then
A
∼
C
.
2.4 Definition
For any positive integer
n
, let
J
n
be the set whose elements are the integers 1
,
2
, . . ., n
; let
J
be the set consisting of all positive integers. For any set
A
, we say:
(a)
A
is
finite
if
A
∼
J
n
for some
n
; the empty set is considered finite.
(b)
A
is
infinite
if
A
is not finite.
(c)
A
is
countable
if
A
∼
J
.
(d)
A
is
uncountable
if
A
is neither finite nor countable.
(e)
A
is
at most countable
if
A
is finite or countable.
2.7 Definition
By a
sequence
, we mean a function
f
defined on the set
J
of all positive integers If
f
(
n
) =
x
n
,
for
n
∈
J
, it is customary to denote the sequence
f
by the symbol
{
x
n
}
, or sometimes by
x
1
, x
2
, x
3
, . . .
.
2.8 Theorem
Every infinite subset of a countable set
A
is countable.
2.9 Definition
Let
A
and Ω be sets, and let
{
E
α
}
be a family of subsets of Ω with index
A
. The
union
of
the sets
E
α
is defined to be the set
S
such that
x
∈
S
if and only if there exists an
α
∈
A
such that
x
∈
E
α
.
We write
S
=
uniondisplay
α
∈
A
E
α
, or
S
=
n
uniondisplay
k
=1
E
n
if
A
=
J
n
, or
S
=
∞
uniondisplay
k
=1
E
n
if
A
=
J.
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 Topology, Metric space, RK, Closed set, General topology

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