Download §2.1. Topological Spaces Let X be a set. A family T of subsets of X is

§2.1. Topological Spaces
Let X be a set. A family T of subsets of X is a topology for
X if T has the following three properties:
(a) Both X and the empty set belong to T .
(b) Any union of sets in T belongs to T .
(c) Any finite intersection of sets in T belongs to T .
A topological space is a pair (X , T ), where X is a set and T
is a topology for X . The sets in T are called open sets.
Example. (a) Let X be a set. Let T0 = {X , ∅}. Then T0 is a
topology for X . It is called the trivial topology. It is the
smallest topology for X in the sense that if T is a topology
for X , then T0 ⊂ T .
(b) Let X be a set. Let T1 be the family of all subsets of X .
Then T1 is a topology for X . It is called the discrete topology.
It is the largest topology for X in the sense that if T is a
topology for X , then T ⊂ T1 .
(c) Let X be a metric space. Then the family of open subsets
(defined in terms of the metric) of X is a topology for X . It is
called the metric topology.
(d) Let Y = {1, 2, 3, . . . }. Let Yn = {1, . . . , n} for each
positive integer n. The family of the sets Yn , together with ∅
and Y , is a topology for Y .
A topological space X is said to be metrizable if the topology
for X is the metric topology associated with some metric on
X.
Example. (a) Let X be a set with more than one element.
Then the topological space X with the trivial topology is not
metrizable. One way to see this is to note that in a metric
space the complement of a singleton needs to be open.
(b) Let X be a set. Then the topological space X with the
discrete topology is metrizable. It is the metric topology
associated with the discrete metric.
Let (X , T ) be a fixed topological space. A subset S of X is
said to be closed if X \ S is open.
1.0. Theorem. (a) Both ∅ and X are closed sets.
(b) Any intersection of closed sets is closed.
(c) A finite union of closed sets is closed.
A subset S of X is said to be a neighborhood of a point x if
there is an open set U such that x ∈ U and U ⊂ S. A point
x ∈ X is said to be an interior point of S if S is a
neighborhood of x. The interior of S is the set of interior
points of S. It is denoted by int(S). It is clear that
int(S) ⊂ S.
1.1. Theorem. A subset S of a topological space is open iff
S = int(S).
1.2. Theorem. If S is a subset of a topological space, then
int(S) is open.
A point x ∈ X is said to be an adherent point of a subset S of
X if S meets every neighborhood of x.The closure of S,
denoted by S, is the set of adherent points of S. Evidently
S ⊂ S. Let S c = X \ S.
1.3. Theorem. Let S be a subset of a topological space X .
c
(a) S = int(S c ).
(b) S is closed iff S = S.
1.4. Theorem. If S is a subset of a topological space X ,
then S is closed.
Example. Let Y be the topological space in the first example
of this section. Let S = {1}. Then S = Y .
A sequence {xj } in a topological space X is said to converge
to a point x ∈ X if for each neighborhood U of x there is an
integer N such that xj ∈ U for all j > N.
Note that a sequence may converge to more than one point.
Example. (a) Let X be a set and let T be the family of
subsets U of X such that X \ U is finite, together with the
empty set. Then T is a topology for X , called the cofinite
topology of X .
(b) Let X have the cofinite topology. Then a sequence
converges to x ∈ X iff each point in X other than x appears in
the sequence at most finitely many times.
(c) Let Z have the cofinite topology. Then the sequence
{1, 2, 3, . . . } converges to each point of Z.
1.5. Theorem. If S is a subset of a topological space X and
if a sequence {xj } in S converges to x ∈ X , then x ∈ S.
The converse of Theorem 1.5 is not true. There may be some
x ∈ S such that no sequence in S converges to x. Recall that
a set is said to be countable if it is empty, or finite, or
denumerable (i.e., having the same cardinality as N, the set of
positive integers).
Example. Let X be an uncountable set and let T be the
family of subsets U of X such that X \ U is countable,
together with the empty set. Then T is a topology for X . A
sequence {xj } in X is convergent iff it is eventually constant.
I.e., {xj } converges to x iff there is an integer N such that
xj = x for all j > N. Let z ∈ X and let S = X \ {z}. Then
S = X . It follows that z ∈ S, but no sequence in S converges
to z.
Let S be a subset of a topological space X . The boundary of
S is ∂S := S ∩ S c . It is clear that ∂S = ∂(S c ). A point x ∈ X
is said to be a boundary point of S if x ∈ ∂S. The exterior of
S is int(S c ).
1.6. Theorem. Let S be a subset of a topological space X .
(a) X is the disjoint union of the interior of S, the boundary of
S, and the exterior of S: X = int(S) ∪ ∂S ∪ int(S c ).
(b) S = int(S) ∪ ∂S, S c = ∂S ∪ int(S c ).
1.7. Theorem. Let S be a subset of a topological space X .
(a) int(S) equals the union of all open sets contained in S.
(b) S equals the intersection of all closed sets containing S.