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Metric and metrizable spaces
These notes discuss the same topic as Sections 20 and 21 of Munkres’ book; some
notions (Symmetric, ∆-metric, Ψ-spaces ...) are not discussed in Munkres’ book.
1. Symmetric, ∆-metric, and metric
Definition 1. Let X be a set. A function d : X 2 → R is called a symmetric on X
if d satisfies the following two axioms:
(1) (∀x, y ∈ X) d(x, y) ≥ 0
and
d(x, y) = 0 iff x = y;
(2) (∀x, y ∈ X) d(x, y) = d(y, x) (the axiom of symmetry.
If, in addition, d satisfies the following axiom of triangle
(3) (∀x, y, z ∈ X) d(x, z) ≤ d(x, y) + d(y, z),
then d is called a metric on X. A function d that satisfies axioms (1) and (3) is
called a ∆-metric.
If d is a metric (symmetric, ∆-metric) on a set X then the pair (X, d) is called
a metric (resp. symmetric, ∆-metric) space.
(1) X =sR; d(x, y) = |x − y| is the standard metric on R;
n
P
(x(k) − y(k))2 is the Euclidean metric on Rn ;1
(2) X = Rn ; d(x, y) =
Examples 2.
k=1
(3) X = Rn ; d(x, y) =
n
P
|x(k) − y(k)| is the NY-metric on Rn ;
k=1
(4) X = Rn ; d(x, y) = max1≤k≤n |x(k) − y(k)| is the maximum metric on Rn ;
(5) Consider the set X from Example 22 in the notes “Countability conditions
and convergence”. For x, y ∈ X, put
 1
 m if one of x, y is bm and the other is c;
1
if one of x, y is am,n and the other is bm ;
d(x, y) = d(y, x) =
 n
1 in all remaining cases.
Then d is a symmetric on X which does not satisfy axiom 3 and thus is not
a metric;
(6) X is any non empty set;
0 if x = y;
d(x, y) =
1 if x 6= y
is a metric on X.
Definition 3. Let X be a set, x ∈ X, d a (sym)metric on X and ε > 0. The ε-ball
around x is
Oε (x) = {y ∈ X : d(x, y) < ε}.
Exercise 4. Sketch ε-balls for Examples 2; in cases 2, 3, 4, take n = 2.
Definition 5. A topology T on the set X is said to be generated by the metric d
on X if the family {Oε (x) : x ∈ X, ε > 0} is a base for T . If a topology T on a
set X can be generated by some metric on X then we say that the space (X, T ) is
metrizable.
A topology T on the set X is said to be generated by the symmetric d on X if
the family if the following two conditions are equivalent for U ⊂ X:
1Here and in the next two examples, n ∈ N, x = hx(1), ..., x(n)i, y = hy(1), ..., y(n)i.
1
2
• U ∈T;
• for every x ∈ U there is ε > 0 such that Oε (x) ⊂ U .
If a topology T on a set X can be generated by some symmetric on X then we say
that the space (X, T ) is symmetrizable.2
Examples 6.
• The metric d from Example 2(6) generates the discrete topology on X.
• The metric d from Example 2(1) generates the standard topology on R.
• The metrics d from Examples 2(2,3,4) all generate the standard topology
on Rn .
• The semymetric d from Example 2(5) generates the topology on X that
we discussed in Example 22 in the notes “Countability conditions and convergence”, and then mentioned again when discussing quotient spaces and
quotient mappings. Recall that this topology is not first-countable.
2. Properties of metrizable spaces; more on countability conditions
Proposition 7. Every metizable space is Hausdorff.3
Proposition 8. Every metizable space is first countable.
Exercise 9. Give an example of a metrizable space which is not second countable.
Definition 10. Let (X, T ) be a topological space. A subset D ⊂ X is called dense
in X if D = X. If X has a dense countable subset then X is called separable.
Definition
11. Let X be a set. A family U of subsets of X is called a cover of X
S
4
if U = X.
A space (X, T ) is called Lindelöf if for every cover U of X consisting of open
sets5 there exists a countable subfamily of U that still covers X.
Theorem 12. Every second countable space is both separable and Lindelöf.
Examples 13.
(1) The Sorgenfrey line is both separable and Lindelöf but (as
we already know) not second countable.
(2) “The one-point Lindelöfication of a discrete space”. Let D be an uncountable set, and p 6∈ D. Put X = D∪{p}. Topologize X declaring the points of
D isolated while a basic neighborhood of p takes the form OC = {p}∪(D\C)
where C is an arbitrary countable subset of D. Then X is Lindelöf but non
separable and thus non second countable.
(3) (R, Tf in ) is separable but non Lindelöf and thus non second countable.
Theorem 14. For a metrizable space X, the following conditions are equivalent:
(1) X is second countable;
(2) X is separable;
(3) X is Lindelöf.6
2The reason for the difference in the definitions is that, in the case of a metric d, the balls
Oε (x) have to be open in the topology generated by d while in the case of a symmetric d they do
not have to.
3In fact, much better than Hausdorff, we will see this later
4Or we say that U covers X
5briefly: for every open cover
6The proof will be discussed in class.
3
Proposition 15. Every subspace of a metrizable space is metrizable.
Proposition 16. Every finite or countable7 product of metrizable spaces is metrizable.
A product of uncountably many factors each of which consists of more than one
point is (not first countable therefore) non metrizable.
3. Properties (and more examples) of symmetrizable spaces
Proposition 17. Every symmetrizable space is T1 .
Proposition 18. Every symmetrizable space is sequential.
Examples 19.
(1) Niemytzky plane (see Hw 3, Part 1, Problem 1) is symmetrizable.
(2) Let’s return to Example 2, (5). This space is symmetrizable, hence sequential, but, as we already know, it is not Fréchet. So a symmetrizable
space
4. Just a little about ∆-metrizable spaces
The Sorgenfrey line is ∆-metrizable8 but not symmetrizable.9
A real life example of a ∆-metric space which is not symmetric: in a city, measure
the distance between points by the time needed to travel; consider a city with some
one-way routes.
Another example: traveling in the mountains.
A real life example of a symmetric space which is not ∆-metric: measure the
distance between cities by the time needed to travel by air. Assume that if there
is a flight from A to B, then there is one from B to A that takes the same time.
Now suppose there is no direct flight from A to C, but there is one from A to B,
and one from B to C, and at B you have to wait.
5. One more generalization of metric spaces; Ψ-spaces
10
Definition 20. A function d : X 2 → R that satisfies axiom 1 from Definition 1 is
called a o-metric on X. Then the pair (X, d) is called an o-metric space. ε-balls are
defined like in Definition 3. If the topology T on X is generated by the o-metric d
in the way described in Definition 5 for symmetrics, then the pair (X, T ) is called
an o-metrizable (or weakly first-countable11) space.
Example 21. The Alexandroff-Urysohn Ψ-space. For each r ∈ P, fix a subset Sr of
Q which can be enumerated as a sequence converging to r. Consider the topology
TΨ on R such that the points of Q are isolated in TΨ while a basic neighborhood of
a point r ∈ P takes the form {r} ∪ (Sr \ F ) where F is an arbitrary finite subset of
Sr . Let us call the pair (R, TΨ ) an AU-Ψ-space.
7Countable products will be discussed later
8Very easy to see
9The proof that I know uses the Baire property (discussed in Chapter 8 of Munkres’ book).
10This section is optional
11Guess why
4
Let E denote the standard topology on R. Then TΨ ⊂ E, so the identity mapping
i : R → R is a condensation of an AU-Ψ-space onto the real line with the standard
topology. Note that Q is an open discrete subspace of (R, TΨ ), and P is a closed
discrete subspace of (R, TΨ ). (R, TΨ ) is first-countable, hence o-mtrizable. With
an accurate choice of the sequences Sr , (R, TΨ ) can be made symmetrizable. Since
(R, TΨ ) contains an uncountable discrete subspace, (R, TΨ ) is not second countable.
Moreover, since (R, TΨ ) contains an uncountable closed discrete subspace, (R, TΨ )
is not even Lindelöf.12 Since (R, TΨ ) is separable but not second countable, (R, TΨ )
is not metrizable.
Example 22. The general definition of a Ψ-space. Let N be a countably infinite
set. A family R of countably infinite subsets of N is called almost disjoint if for
every two distinct r1 , r2 ∈ R, the intersection r1 ∩ r2 is finite.13 So let R be an
almost disjoint family of infinite subsets of N . The set Ψ(R) = N ∪R is topologized
by declaring the points of N isolated while a basic neighborhood of r ∈ R takes
the form {r} ∪ (r \ F ) where F is an arbitrary finite subset of r.14
A “general” Ψ space has properties similar to those of the AU-Ψ-space: N is open
and discrete in Ψ(R) while R is closed and discrete. Ψ(R) is always first countable.
Ψ(R) is second countable iff R is countable (and in this case Ψ(R) is homeomorphic
to the discrete union of countably many copies of the space “convergent sequence”).
Definition 23. A function f : X → R is bounded if there is a constant M such
that |f (x)| < M for every x ∈ X. A topological space is pseudocompact if every
continuous function f : X → R is bounded.
Definition 24. Let N be a countably infinite set. An almost disjoint family R
consisting of infinite subsets of N is called maximal almost disjoint15 if R is not a
proper subset of any larger almost disjoint family consisting of infinite subsets of
N.
One can prove using the axiom of Choice that every almost disjoint family can
be extended to a maximal almost disjoint family.
Proposition 25. Ψ(R) is pseudocompact iff R is a mad family.
Ψ(R) with a mad family R is not symmetrizable except when R is finite.
6. Bounded metrics
It is natural to call a metric d on a set X bounded if there is a constant M such
that d(x, y) < M for every x, y ∈ X.
Notation 26. Let d be a metric on a set X. For every x, y ∈ X, put
d(x, y) = min{d(x, y), 1}.
Then d is called the standard bounded metric corresponding to d.
12It is a good exercise to show that a closed discrete subspace of a Lindelöf space can be at
most countable.
13Thus the family {S : r ∈ P} in Example 21 is almost disjoint because sequences converging
r
to different points may have only finitely many common points. This also shows that an almost
disjoint family of subsets of a countable set may have cardinality as big as c (while a disjoint
family of subsets of a countable space is at most contable).
14So r ∈ R is, at the same time, a point of Ψ(R), and a subset of Ψ(R).
15Abbreviated as mad
5
Proposition 27. (1) d is a metric.
(2) The topology on X generated by d is the same as the topology generated by d.
So every metrizable topology can be generated by a bounded metric. Here is
another way:
˜ y) =
Exercise 28. Let d be a metric on a set X. For every x, y ∈ X, put d(x,
˜
˜
Arctan d(x, y). Show that d is a metric on X, and that d generates the same
topology on X as d.
7. Metrics on finite products
Examples 29. Let n ∈ N and for each k ∈ {1, ..., n} let (Xk , Tk ) be a topological space and dk a metric on Xk thatQgenerates the topology Tk . For x =
hx(1), ..., x(n)i, y = hy(1), ..., y(n)i ∈ P =
Xi put
1≤i≤n
v
u n
uX
dEuclidean (x, y) = t (x(k) − y(k))2 ;
k=1
dNY (x, y) =
n
X
|x(k) − y(k)|;
dmax (x, y) = max |x(k) − y(k)|.
1≤k≤n
k=1
Then each of dEuclidean , dNY , dmax (is a metric and) generates the Tychonoff product
topology on P .
8. Metrics on infinite products
Definition 30. Let for each a ∈ A, (Xa , Ta ) be a topological
Q space such that the
topology Ta is generated by the metric da . For x, y ∈ P =
Xa , put
a∈A
d(x, y) = sup da (x(a), y(a)).
a∈A
Then d is called the uniform metric on P and the topology generated by d is called
the uniform topology.
The uniform topology on P is stronger than the Tychonoff product topology but
weaker than the box product topology.
Proposition 31. Let for each n ∈ N, (Xn , Tn ) be a topological
Q space such that the
topology Tn is generated by the metric dn . For x, y ∈ P =
Xn , put
n∈N
1
dn (x(n), y(n)).
n
n∈N
D(x, y) = sup
Then D generates the Tychonoff product topology on P .
9. Mappings of metric spaces
Proposition 32. Let (X, dX ) and (Y, dY ) be two metric spaces. A mapping f :
X → Y is continuous (with respect to the topologies generated on X and Y by dX
and dY ) iff for every x0 ∈ X and every ε > 0 there exists δ > 0 such that for every
x ∈ X, if dX (x, x0 ) < δ then dY (f (x), f (x0 )) < ε.
6
Definition 33. Let (X, dX ) and (Y, dY ) be two metric spaces. A mapping f :
X → Y is called uniformly continuous (with respect to the metrics dX and dY ) if
for every ε > 0 there exists δ > 0 such that for every x0 , x ∈ X, if dX (x, x0 ) < δ
then dY (f (x), f (x0 )) < ε.
Example 34. The mapping f : R → R is continuous (with respect to the standard
topology on R) but not uniformly continuous (with respect to the standard metric
on R.
Definition 35. Let (X, T ) be a topological space. A sequence of functions {fn :
n ∈ N} (where fn : X → R) converges pointwise to a function f : X → R if for
every x ∈ X, lim fn (x) = f (x) (that is for every x ∈ X for every ε > 0 there is
n→∞
N (ε, x) such that for all n > N (ε, x), |fn (x) − f (x)| < ε). In this case we say that
f is the pointwise limit of the sequence {fn : n ∈ N}.
A sequence of functions {fn : n ∈ N} (where fn : X → R) converges uniformly
to a function f : X → R if for every ε > 0 there is N (ε) such that for all n > N (ε),
for every x ∈ X, |fn (x) − f (x)| < ε).In this case we say that f is the uniform limit
of the sequence {fn : n ∈ N}.
Definition 35 corresponds to the convergence in the sense of pointwise or uniform
topology on RX .16
Example 36. The pointwise limit of a sequence of continuous functions is not
necessarily continuous.
Indeed, put

 0 if x ≤ 0;
nx if 0 ≤ x ≤
fn (x) =

1 if x ≥ n1
1
n;
for n ∈ N
and f (x) =
0 if x ≤ 0;
1 if x > 0.
Proposition 37. The uniform limit of a sequence of continuous functions is a
continuous function.
16Note that typically RX is not sequential, so convergent are not enough to describe its
topology.