§ 8. Topological spaces.
8.1. Definition. A pair (X, T ), where T ⊂ 2
X, is called a topological space,
if T satisfies the following conditions:
(1) ∅, X ∈ T ;
(2) U1, U2 ∈ T ⇒ U1 ∩ U2 ∈ T ;
(3) the union S
U∈T0
U of an arbitrary family T0 ⊂ T belongs to T .
T is said to be a topology on the set X. Usually the topological space
(X, T ) is denoted by X, if it is clear that a set with a topology is being considered. The elements of
the set X are called points of the space X.
The elements of the topology T are called open sets of the space X, and their complements — closed sets.
From properties (1), (2) and (3) the following properties follow:
(1’) ∅ and X are closed sets of the space X;
(2’) the union of a finite number of closed sets is closed;
(3’) the intersection of any family of closed sets is closed.
The family of all topologies on a set X is ordered by the inclusion relation:
T1 ≤ T2 ⇐⇒ T1 ⊂ T2,
i.e., every set U open in the topology T1 is also open in the topology T2.
In this case one says that the topology T1 is weaker than the topology T2, and the topology T2 is stronger than
T1. From Definition 8.1 it follows that the pair {∅, X} is contained in every topology T on X.
It is also clear that this pair is a topology on X, hence {∅, X} is the smallest or
weakest topology on X. It is called the antidiscrete topology.
The family 2
X of all subsets of the set X is also a topology on X. From Definition 8.1 it follows that this is the largest or strongest topology on X. It is called
the discrete topology. In a discrete space every set is simultaneously open and closed.
8.2. Examples.
1) The simplest examples of topological spaces are X = ∅ and a set X consisting of a single point a. The only topology on these sets is the pair {∅, X}.
2) On a set X consisting of two distinct points a and b, there are four different
topologies.
T1 = {∅, X} — the glued two-point space;
T2 =
∅, {a}, X
— the connected two-point space;
T3 =
∅, {b}, X
— the connected two-point space;
T4 =
∅, {a}, {b}, X
— the discrete two-point space.
3) A more complicated situation arises in the case of a set X consisting of n distinct
points. From Definition 8.1 it follows that the number of distinct topologies on this set does not
exceed 2
2
n
.
4) On a set X consider the family F of all its finite subsets together with the set
X itself. This family is closed under finite unions and arbitrary intersections. Hence
the family F defines a topology through the definition of the closed subsets of X. It is called
the topology of finite complements.
5) Let T be a topology on X and Y ⊂ X. Then, as is easily seen (using the distributive law in the proof of condition (3) of Definition 8.1), the family
T |Y = {U ∩ Y : U ∈ T }
is a topology on the set Y . The space (Y, T |Y ) is called a subspace of the space (X, T ). One usually says: “Y is a subspace of the space X,” T |Y being the topology
induced by the topology T on Y .
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8.3. Definition. A family B of open subsets of a space X is called an open base of X, if every open set of the space X is a union of some
elements of B.
8.4. Proposition. Let B be a family of subsets of a set X satisfying the
conditions:
(1) every point x ∈ X belongs to some element U ∈ B;
(2) if x ∈ U1∩U2 and U1, U2 ∈ B, then there exists an element U3 ∈ B such that x ∈ U3 ⊂ U1∩U2.
Then B is a base of some (uniquely determined) topology on the set X.
Proof. Set
T =
n [
O∈B0
O : B0 ⊂ Bo
(8.1)
and let us show that T is a topology on X. The empty set belongs to T , since
∅ =
S
O∈B0
O for B0 = ∅ ⊂ B. Further, by condition (1) X =
S
O∈B O ∈ T . Condition (3)
of Definition 8.1 is satisfied automatically. Now let us show that the intersection of two elements
U1 and U2 of T belongs to T .
Let U1 =
S
V ∈B1
V, U2 =
S
V 0∈B2
V
0
. Then
U1 ∩ U2 =
S
V ∈B1
V
∩
S
V 0∈B2
V
0
=
S
V ∈B1
S
V 0∈B2
(V ∩ V
0
).
But every set of the form V ∩V
0
is, by condition (2), a union of sets W ∈ B.
Thus, the intersection U1 ∩ U2 is also a union of sets W ∈ B. Consequently,
T satisfies condition (2) of Definition 8.1.
Finally, if the family B is a base of some topology, then this topology is uniquely
determined. But we have just proved that the family T from (8.1) is a topology with base B.
8.5. Examples of topologies on R.
1) T1 — the topology on R whose base consists of the intervals (a, b), a < b, a, b ∈ R (the standard
topology).
2) T2 — the topology of finite complements. The topology of finite complements on the line R is called the Zariski topology.
3) T3 — the topology on R whose base consists of the half-intervals [a, b), a < b, a, b ∈ R (the line
with this topology is called the Sorgenfrey line).
4) T4 — the topology on R whose base consists of the rays (a, ∞) = {x ∈ R : a < x}, a ∈ R.
5) Let K = {
1
n
: n ∈ N}. T5 — the topology on R whose base consists of the sets (a, b),
(a, b) \ K, a < b, a, b ∈ R.
8.6. Definition. A family P of open subsets of a space X is called an
open subbase of X, if the set of all possible finite intersections U1∩. . .∩Uk of elements
Ui ∈ P is a base of the space X.
A family v of subsets of a set X is called a cover of the set X, if X = S
V ∈v
V .
8.7. Proposition. Let X be a set and v an arbitrary cover of it. Then v
is a subbase of some uniquely determined topology on the set X.
The proof reduces to the fact that the family B of all possible finite intersections of
elements of v satisfies the conditions of Proposition 8.4.
§ 9. Topology of a linear order.
9.1. Definition. Let (X, <) be a linearly ordered set with more than one point. The family of sets P:
1) (−∞, b) = {t ∈ X : t < b}, a ∈ X,
2) (a, +∞) = {t ∈ X : a < t}, a ∈ X
form a subbase of the order topology on X.
We call the topology from Definition 9.1 the topology generated by the linear order,
or the topology of the linear order. The sets (a, b) = {t ∈ X : a < t < b}, a < b, a, b ∈ X
are called open intervals.
9.2. Example. On the plane R
2
with the lexicographic order, the open intervals
((x1, y1),(x2, y2)) are:
1) vertical intervals ((x, y1),(x, y2)), if x = x1 = x2, y1 < y2,
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2) the union of the vertical ray {(x1, t) : y1 < t}, the vertical lines x = t, x1 < t < x2
and the vertical ray {(x2, t) : t < y2}, if x1 < x2.
9.3. Proposition. Let (X, <) be a linearly ordered set and Y ⊂ X. If T
is the order topology on (X, <), T
0 the order topology on (Y, < |Y ), then T
0 need not
coincide with T |Y .
The proof reduces to constructing an example. X = R, Y = (0, 1) ∪ {2}. Then {2} is an
open subset of (Y, T |Y ), while the neighborhoods of the point {2} in the space (Y, T
0
) have the form
(a, 1) ∪ {2}, where 0 < a < 1.
§ 10. Metric spaces.
10.1. Definition. A metric space is a pair (X, ρ), where X is a set, and ρ is a metric on X, i.e., a nonnegative function
ρ : X × X → R+,
satisfying the following conditions:
(1) ρ(x, y) = 0 ⇐⇒ x = y (identity axiom);
(2) ∀x, y ∈ X ρ(x, y) = ρ(y, x) (symmetry axiom);
(3) ∀x, y, z ∈ X ρ(x, z) ≤ ρ(x, y) + ρ(y, z) (triangle axiom).
For brevity, a metric space (X, ρ) is often denoted by a single letter X. Elements of a metric space are called points. The number ρ(x, y) is called the distance
between the points x and y.
10.2. Examples of metric spaces.
1. The simplest examples are the empty set and a set X consisting of a single
point. On each of these sets there exists a unique metric.
2. Euclidean spaces. After the introduction of rectangular coordinates the spaces En,
n ∈ N, turn into arithmetic n-dimensional spaces R
n, in which the distance
between points x = (x1, . . . , xn) and y = (y1, . . . , yn) is computed by the formula
ρ(x, y) =
vuutXn
i=1
(xi − yi)
2. (10.1)
3. Any set X turns into a metric space if we set ρ(x, y) = 1
for any distinct points x, y ∈ X.
4. Let (X, ρ) be a metric space and Y ⊂ X. Define on Y the metric ρ|Y as
the restriction of the metric ρ to Y , i.e.
ρ|Y
(x, y) = ρ(x, y) for any x, y ∈ Y .
It is clear that the distance function ρ|Y defined in this way satisfies the axioms of a metric. Formally one should write not “ρ|Y ”, but “ρ|Y ×Y ”, but for convenience we use the shorter notation.
The pair (Y, ρ|Y ) is called a subspace of the metric space (X, ρ). Subspaces give us a large supply of metric spaces.
The following examples of metric spaces are related to the concept of a normed space, known from the course “Linear Algebra and Geometry.” Let us recall the definition in the simplest
case.
10.3. Definition. Let V be a real linear space. A norm on the space
V is a mapping
|| · || : V → R+,
assigning to a vector x ∈ V a nonnegative number ||x|| and satisfying the axioms
(1) if ||x|| = 0, then x = 0;
(2) ||αx|| = |α| · ||x|| for every α ∈ R and ∀x ∈ V ;
(3) ∀x, y ∈ V ||x + y|| ≤ ||x|| + ||y|| (triangle axiom).
From (2) it follows that ||0|| = 0.
A linear space V with a norm || · || defined on it is called a normed (linear) space.
10.4. Examples of normed spaces.
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1. The simplest example is the zero-dimensional space V = {0}, on which there exists
a unique norm.
2. Norms on the arithmetic n-dimensional space R
n, whose vectors are sequences x = (x1, . . . , xn) of real numbers:
||x|| = |x1| + . . . + |xn| (10.2)
(the norm axioms are checked using the properties of the (real) modulus);
||x|| =
vuutXn
i=1
x
2
i
(10.3)
(in the course “Linear Algebra and Geometry” it is proved that the function from (10.3) satisfies all the
norm axioms);
||x|| = max
|x1|, . . . , |xn|
(10.4)
(the norm axioms are checked coordinatewise using the properties of the (real) modulus
of a number).
On R (in the one-dimensional space) all the norms (10.2)–(10.4) coincide and are defined as follows:
||x|| = |x|.
3. The norm (10.4) from example 2 is generalized to a norm on the space C
[0, 1], R
of continuous
real functions on the segment [0, 1] ⊂ R:
||f|| = sup
|f(t)| : t ∈ [0, 1]
.
By the Weierstrass theorem the norm is well defined. Let us check the triangle axiom. Let
f = g + h. Then for every t ∈ [0, 1] we have |f(t)| ≤ |g(t)| + |h(t)|. From this it follows that
|f(t)| ≤ sup
|g(t
0
)| : t
0 ∈ [0, 1]
+ sup
|g(t
0
)| : t
0 ∈ [0, 1]
,
whence we obtain the inequality ||f|| ≤ ||g|| + ||h||.
10.5. Proposition. Let (V, ||·||) be a normed space. For x, y ∈ V set
ρ(x, y) = ||x − y||. (10.5)
Then ρ is a metric on the set V .
Proof. It is clear that it suffices to check the triangle axiom (3) in Definition
10.1. We have
ρ(x, z) = (10.5) = ||x − z|| = ||x − y + y − z|| ≤
by the triangle axiom (10.3)
≤ ||x − y|| + ||y − z|| = ρ(x, y) + ρ(y, z).
Thus, Proposition 10.5 gives us new examples of metric spaces: the spaces R
n, n ≥ 2, with three distinct metrics (the norms (10.2)–(10.4) are distinct for n ≥ 2
and, consequently, lead to different metrics).
§ 11. Topology of a metric space. Metrizable spaces.
11.1. Definition. Let (X, ρ) be a metric space, x ∈ X and ε > 0. The set
Oε(x) =
x
0 ∈ X : ρ(x, x0
) < ε
is called the ε-neighborhood of the point x (or the open ε-ball centered at the point x) in the metric space X.
11.2. Proposition. The set of all ε-neighborhoods Oε(x), ε > 0, of points x of a metric
space (X, ρ) forms a base of some topology on X.
Proof. It suffices to show that the set
B =
Oε(x) : ε > 0, x ∈ X
satisfies the conditions of Proposition 8.4. Condition (1) is satisfied obviously. Now let
x ∈ Oε1
(x1) ∩ Oε2
(x2).
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Then
ri = ρ(x, xi) < εi
, i = 1, 2. (11.1)
Set
ε = min{ε1 − r1, ε2 − r2}.
Condition (11.1) implies that ε > 0, and from the triangle axiom we obtain
Oε(x) ⊂ Oε1
(x1) ∩ Oε2
(x2).
Thus, condition (2) of Proposition 8.4 is also satisfied.
Proposition 11.2 can be rephrased as follows.
11.3. Proposition. A set U is open in the metric topology Tρ, if for every
point x ∈ U there is an open ε-ball Oε(x) centered at x, contained in U.
The topology from Proposition 11.2 will be denoted by Tρ. It is called the topology generated by the metric ρ,
or the metric topology.
11.4. Examples. The base of the metric topology Tρ of the spaces R
n from Example 10.2.2 (open ε-balls) is formed by:
1) intervals (x − ε, x + ε) of length 2ε centered at points x of the line R;
2) disks (without boundary circles) of radius ε centered at points x of the plane R
2
;
3) balls (without boundary spheres) of radius ε centered at points x of the space R
3
.
11.5. Proposition. If (X, ρ) is a metric space and Y ⊂ X, then the topologies
Tρ|Y and Tρ|Y
on Y coincide.
The proof reduces to a simple verification that for every point y ∈ Y the set
Oε(y) ∩ Y coincides with the ε-neighborhood of the point y in the metric ρ|Y .
Metric spaces give us a large supply of topological spaces, but not every
topology is generated by a metric.
11.6. Definition. A topological space X is called metrizable, if on X
there exists a metric ρ such that the metric topology Tρ coincides with the topology of the space
X.
The topologies T2 and T3 from Example 8.2 (2) are not metric topologies. The space
connected two-point space is not a metrizable space.
Problem Set No. 2
1. Describe the bases of the topology of the discrete and antidiscrete spaces.
2. Let Y = [−1, 1] be a subspace of R with the standard topology. Which of the subsets Y
are open in Y, open in R: 1) {x :
1
2 < |x| < 1}, 2) {x :
1
2 ≤ |x| < 1}, 3) {x :
1
2 < |x| ≤ 1}, 4)
{x :
1
2 ≤ |x| ≤ 1}, 5) {x : 0 < |x| < 1,
1
x
6∈ N}?
3. Find the least upper bound and greatest lower bound of the topologies T1 = {∅, X, {a}, {a, b}} and T2 =
{∅, X, {a}, {b, c}} on X = {a, b, c}.
4. Prove:
1) the topologies in Example 8.5 of Lecture 2 are well defined,
2) T4 ≤ T1 ≤ T5, T2 ≤ T1 ≤ T3,
3) T2 and T4 are incomparable, T5 and T3 are incomparable.
4) find the least upper bound and greatest lower bound of the given topologies.
5. Prove that any open subset of the line is a union of at most
countably many disjoint intervals (the whole line and the
open rays (∞, a), (a, ∞) are additionally considered intervals).
What is the cardinality of the standard topology of the line?
6. Can different topologies on a set X induce the same topologies on a
subset A ⊂ X?
7. Let Y be a subset of X. Prove that a subset F is closed in Y if and only
if there exists a closed subset Φ in X such that F = Φ ∩ Y .
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8. Let Y be an open (closed) subset of X. Prove that any open (closed) subset of Y is open (closed) in X.
9. Is the topology of the lexicographic order on the square N×N of the set of
natural numbers N with the natural well-ordering discrete?
10. Prove that the topology on the set [0, 1] × [0, 1], induced by the topology of the lexicographic order on the square R × R of the set of real numbers R, and the topology
of the lexicographic order on the square [0, 1] × [0, 1] of the segment [0, 1] are different.
11. Let a rectangular coordinate system be given on the plane R
2
, s = (xs, ys), q = (xq, yq).
Define the mappings ρ? : R × R → R+:
(ρd) ρd(s, q) = 1, if s 6= q, ρd(s, q) = 0, if s = q;
(ρ1) ρ1(s, q) = |xs − xq| + |ys − yq|;
(ρ2) ρ2(s, q) = p
|xs − xq|
2 + |ys − yq|
2;
(ρ∞) ρ∞(s, q) = max{|xs − xq|, |ys − yq|};
(ρj ) ρj (s, q) = |ys − yq|, if xs = xq, ρj (s, q) = |xs − xq| + |ys| + |yq|, if xs 6= xq.
Verify that they are metrics. Draw the unit open balls of points in these
metrics. Compare the topologies generated by these metrics.
12. Let
(ρ1) `
1 = {x = (xi) : P∞
i=1 |xi
| < ∞} — the space of sequences of real
numbers with the norm ||x||1 =
P∞
i=1 |xi
|;
(ρ2) `
2 = {x = (xi) : P∞
i=1 |xi
|
2 < ∞} — the space of sequences of real
numbers with the norm ||x||2 = (P∞
i=1 |xi
|
2
)
1
2 ;
(ρ∞) `∞ = {x = (xi) : supi∈N |xi
| < ∞} — the space of sequences of real
numbers with the norm ||x||∞ = supi∈N |xi
|.
Verify that the normed spaces are well defined. Write out the formulas for the
metrics defined by these norms.
13. Prove that on the space C
[0, 1], R
of continuous real functions on the segment [0, 1] the norms
(ρ1) ||f||1 =
R
1
0
|f(x)|dx;
(ρ2) ||f||2 = (R
1
0
|f(x)|
2dx)
1
2 ;
(ρ∞) ||f||∞ = sup{|f(x)| : x ∈ [0, 1]}
are well defined. Write out the formulas for the metrics defined by these norms. Compare
the topologies generated by these norms.
14. Let (X, ρ) be a metric space. Prove that the mappings ρ1 and ρ2 : X × X →
R+, given by the formulas ρ1(x, y) = min{1, ρ(x, y)} and ρ2(x, y) = ρ(x, y)/(1 + ρ(x, y)), are
metrics on the set X. Compare the topologies generated by the metrics ρ, ρ1 and ρ2.
15. Can an open ball of larger radius in a metric space be contained in an
open ball of smaller radius?
Additional Problems for Problem Set No. 2
16. Let a family of topologies A be given on a set X. Prove that there exist: the least
upper bound of the topologies from A (the smallest topology greater than any topology from A); the greatest
lower bound of the topologies from A (the largest topology smaller than any topology from A).
17. Describe the open subsets of the line in the topologies from problem 4. What are the cardinalities of their
topologies?
18. Give an example of a topological space for which there exists a subbase of the topology whose cardinality is less than the cardinality of any of its base.
Is it true that on a finite discrete space of n points the number of elements in any
subbase is at least n?
Prove that if the cardinality of any base of a topological space is infinite (additionally ≥ κ), then the cardinality of any subbase is also infinite (additionally ≥ κ).
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19. Let p be a prime number and the difference x − y of distinct numbers x, y ∈ Q be represented in the form
r
s
p
α, where r, s and α ∈ Z, r, s coprime to p. Set ρ(x, y) = p
−α for x 6= y, x, y ∈ Q.
Prove that ρ is a metric (the p-adic metric on Q). Compare the Euclidean topology on the
set of rational numbers Q and the topology of Q generated by the p-adic metric on Q.
20. The Baire space. Let X be an arbitrary infinite set. On the set XN
introduce a metric ρ as follows: for p = (x1, x2, ...), q = (y1, y2, ...) ∈ B set ρ(p, q) = 0,
if p = q, and ρ(p, q) = 1/k, if k is the smallest natural number for which xk 6= yk.
Show that the metric is well defined. Give a description of the open balls.
21. "Metrizable hedgehog." Let Λ be some infinite set. Assign to
each element λ ∈ Λ a segment [0, 1], which we denote by [0, 1]λ and assume that all
these segments are pairwise disjoint except for the point 0, which is assumed to
belong to all the segments. Set X = ∪{[0, 1]λ : λ ∈ Λ} and define a metric ρ :
X × X → R+. For p, q ∈ X set ρ(p, q) = |p − q|, if p and q belong to the same segment
[0, 1]λ for some λ ∈ Λ, and ρ(p, q) = p + q, if p and q do not belong to the same segment [0, 1]λ.
Show that the metric is well defined. Give a description of the open balls.
22. Let (X, ρ) be a metric space and M ⊂ X. If sup
ρ(x, y) : x, y ∈ M
= d < ∞,
then the set M is called bounded, and the number d is called its diameter and is denoted diam(M). Prove that for any metric space (X, ρ) the Hausdorff
distance
dρ(A, B) = max{sup{ρ(a, B) : a ∈ A},sup{ρ(b, A) : b ∈ B}}
is a metric on the set of bounded closed subsets A, B ⊂ X. Can one
drop the requirement that they be closed? bounded?
23. Prove the metrizability of the topology of the lexicographic order on the square R × R of the linearly ordered set R. Give an example of a metric inducing this topology.
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