The Stone–Weierstrass theorem. Compact-open topology on C(X, Y ). Maps X × Y → Z and X → C(Y, Z). Connectedness. Connected components.
40.4. Theorem (Stone–Weierstrass). Let a ring K of continuous real-valued functions on a Hausdorff compact space X contain all constant functions, separate points, and be closed with respect to the topology of uniform convergence on C
∗
(X).
Then K = C
∗
(X).
Proof. For each function k ∈ K other than a constant, set g(x) =
k(x)+Mk
2Mk
, where Mk = ||k||. Denote the family of functions g : X → I by A. The diagonal product
F = ∆{g : g ∈ A} : X → I
A,
being an injective continuous map of a compact space, is an embedding of X
into the Tychonoff cube I
A.
Any function f : X → R extends, by the Brouwer–Tietze–Urysohn Theorem, to a function
˜f : I
A → R such that f = ˜f ◦ F. By Theorem 40.3, for any ε > 0 there exists a polynomial
function on I
A
p(y) = p
1
1
(prg
1
1
(y)) · · · · · p
1
k(1)(prg
1
k(1)
(y)) + · · · + p
m
1
(prgm1
(y)) · · · · · p
m
k(m)
(prgm
k(m)
(y))
such that |
˜f(y) − p(y)| < ε, for any point y ∈ I
A. Then for any point x ∈ X we have
|f(x) − p(F(x))| = |
˜f(F(x)) − p(F(x))| < ε.
At the same time, since g = prg ◦ F for g ∈ A, we have
p(F(x)) = p
1
1
(prg
1
1
(F(x)))·· · ··p
1
k(1)(prg
1
k(1)
(F(x)))+· · ·+p
m
1
(prgm1
(F(x)))·· · ··p
m
k(m)
(prgm
k(m)
(F(x))) =
= p
1
1
(g
1
1
(x)) · · · · · p
1
k(1)(g
1
k(1)(x)) + · · · + p
m
1
(g
m
1
(x)) · · · · · p
m
k(m)
(g
m
k(m)
(x)).
The functions p
j
i
◦ g
j
i ∈ K, j = 1, . . . , m, i = 1, . . . , k(j), hence p ◦ F ∈ K also.
§ 41. Compact-open topology on C(X, Y ).
41.1. Compact-open topology on C(X, Y ). A subbase of the compact-open topology Tco on the set C(X, Y ) is formed by sets of the form
[K, U] = {f ∈ C(X, Y ) : f(K) ⊂ U},
where K is a compact set in X, and U an open set in Y .
41.2. Proposition. If the space Y is Hausdorff, then the space (C(X, Y ), Tco)
is also Hausdorff.
Proof. Let fi
: X → Y be distinct maps, i = 1, 2. There exists a point
x ∈ X, for which f1(x) 6= f2(x). Take disjoint neighborhoods U1 and U2 of the points f1(x)
and f2(x). Then the sets Vi =
{x}, Ui
, i = 1, 2, do not intersect and fi ∈ Vi
.
41.3. Theorem.
Tco ≥ T |C(X,Y )
.
If X is discrete, then Tco = T |C(X,Y )
.
If (Y, d) is a metric space, then
Td0 |C(X,Y ) ≥ Tco,
and the topologies coincide in the case of a Hausdorff compact space X.
Proof. The set U = pr−1
x O ∩ C(X, Y ), x ∈ X, from the subbase of the topology T |C(X,Y )
is obviously a set of the form [{x}, O] from the subbase of the compact-open topology Tco.
Thus the first inequality is proved. Equality of the topologies in the case of discrete X follows
from the finiteness of compact subsets of X.
To prove the second inequality it suffices to show that any set [K, U] from the
subbase of the compact-open topology is open in the topology of uniform convergence. Let
f = (f(x))x∈X ∈ [K, U]. Since f(K) ⊂ U and f(K) is compact, d(f(K), Y \U) = inf{d(t, Y \U) :
t ∈ f(K)} = > 0 (the function dY \U : X → R+, dY \U (x) = d(x, Y \ U), is continuous (Example 13.6
of Lecture 3, and f(K) is compact). Then O(f) ⊂ [K, U] and the second inequality is proved.
70
Let us prove equality of the topologies in the case of a Hausdorff compact X. For this it suffices
to show that any neighborhood of an arbitrary map in the topology of uniform convergence contains a neighborhood of it in the compact-open topology.
Take a basic neighborhood O(f) of a map f in the topology of uniform convergence.
For any point x ∈ X there exists its neighborhood Ox such that f(Cl(Ox)) ⊂ Oε
2
(f(x)). By virtue of
the compactness of X, choose a finite subcover {Ox1
, . . . , Oxn } from the cover {Ox : x ∈
X}. Then for the sets [Cl(Oxi
), Oε
2
(f(xi))], i = 1, . . . , n, from the subbase of the compact-open
topology, we have
f ∈
\n
i=1
[Cl(Oxi
), Oε
2
(f(xi))] ⊂ O(f).
§ 42. Maps X × Y → Z and X → C(Y, Z).
42.1. Maps X × Y → Z and X → C(Y, Z). Let spaces X, Y, Z and a continuous map f : X × Y → Z be given. Then there is defined a map F : X → C(Y, Z),
(F(x))(y) = f(x, y). (42.1)
Now let a map F : X → C(Y, Z) be given. Then there is defined a map f : X ×Y → Z,
given by the formula
f(x, y) = (F(x))(y). (42.2)
42.2. Theorem. For a continuous map f : X × Y → Z the map F : X →
C(Y, Z) (42.1), into the space of continuous maps C(Y, Z) with the compact-open topology, is continuous.
The converse statement holds if the space Y is locally compact.
Proof. By Proposition 13.3 of Lecture 3 it suffices, for a point x0 ∈ X and a
subbasic neighborhood [K, U] of the function F(x0)
(i.e. (F(x0))(K) ⊂ U) (42.3)
to find a neighborhood Ox0 such that F(Ox0) ⊂ [K, U].
For each point y ∈ K fix neighborhoods Vy and Wy of the points x0 in X and y in Y
respectively, such that
f(Vy × Wy) ⊂ U. (42.4)
This is possible, by virtue of (42.1) and (42.3). From the family {Wy : y ∈ K} select a finite subfamily
Wy1
, . . . , Wyn
, covering the set K. Then Ox0 =
Tn
i=1 Vyi is a neighborhood of the point
x0 and
f(Ox0 × K) ⊂ f(
[n
i=1
Vyi × Wyi
) ⊂ (42.4) ⊂ U. (42.5)
From (42.5) and (42.1) it follows that F(Ox0) ⊂ [K, U].
Proof of the converse statement. For a point (x0, y0) ∈ X × Y and a neighborhood W of
f(x0, y0), take a neighborhood V of y0 with compact closure Cl(V ), contained in
(F(x0))−1
(W). By virtue of the continuity of the map F, there exists a neighborhood U of x0,
such that F(U) ⊂
Cl(U), W
. Then U × W is a neighborhood of the point (x0, y0), satisfying the condition
f(U × V ) ⊂ W by the definition (42.2) and the condition F(U) ⊂
Cl(V ), W
.
§ 43. Connectedness. Connected components.
43.1. Definition. A space X is called connected if it cannot be represented as
the union of two nonempty disjoint open sets.
A space X is disconnected if and only if X can be represented as X = A ∪ B,
a union of disjoint nonempty clopen sets A and B.
43.2. Examples. 1. The empty set and a space consisting of a single point are connected.
2. If a space X consists of two distinct points a and b (Example 8.2.2 of Lecture 2), then
the topologies T1 = {∅, X} (the "glued two-point space"), T2 =
∅, {a}, X
, T3 =
∅, {b}, X
(the "connected
two-point spaces") are connected.
3. A finite T1-space containing at least two points is disconnected.
4. The subspaces Q ⊂ R (respectively P ⊂ R) of rational (respectively irrational) numbers of the number line R are disconnected.
5. The segment [0, 1] is connected (proved in the calculus course).
71
43.3. Lemma. If a connected subspace X0 of a space X intersects a clopen set U ⊂ X, then X0 ⊂ U.
Proof. Indeed, otherwise X0 = (U ∩ X0) ∪ ((X \ U) ∩ X0), where
(U ∩X0) and (X \U)∩X0 are nonempty disjoint open subsets of X0. This contradicts the
connectedness of X0.
43.4. Proposition. Let Zα be connected subsets of a space X, α ∈ A. If
Z0 =
T
α∈A Zα 6= ∅, then the set Z =
S
α∈A Zα is connected.
The closure of a connected set is connected.
Proof. If the first statement is false, then Z is represented as the union of disjoint nonempty clopen sets A and B. Let the point x0 ∈ Z0
belong to one of these sets, say, x0 ∈ A. Then for any point x ∈ Z the connected set Zα, to which the points x0, x belong, belongs to A by Lemma 43.3.
Hence Z ⊂ A and B = ∅. We have a contradiction.
Let Z be a connected subset of a space X. If the second statement is false, then Cl(Z)
is represented as the union of disjoint nonempty clopen sets
A and B. Since Z is dense in Cl(Z), A ∩ Z 6= ∅ and B ∩ Z 6= ∅. Hence Z is disconnected.
We have a contradiction.
43.5. Properties of connectedness.
1. Let Z be a connected subset of X. Then any subset Z
0 such that Z ⊂ Z
0 ⊂
Cl(Z), is connected.
2. The continuous image of a connected space is connected.
3. The Tychonoff product of connected spaces is a connected space.
Let X =
Q
α∈A Xα and Xα be a connected space, α ∈ A. Fix a point x = (xα) ∈ X
and for any finite subset F = {α1, . . . , αk} ⊂ A consider the subset XF =
{y = (yα) ∈ X : yα = xα, if α 6∈ F}. It is easy to see that XF is homeomorphic to the product
Xα1 × . . . × Xαk
.
Denote the family of finite subsets of A by AFin, Y =
S
F ∈AFin XF . Then Y is dense
in X. Indeed, any basic open set has the form O =
Tk
i=1 pr−1
αi Oi
,
where Oi is open in Xαi
. Then for F = {α1, . . . , αk} ⊂ A the point (yα), yα = xα, if α 6∈ F, and
yαi ∈ Oi
, is contained in O ∩ XF . Hence, if Y is connected, then X is also connected (as its closure).
The subset Y is connected if all subsets XF , F ∈ AFin, are connected (their intersection is
nonempty (the point x belongs to all these sets)).
The connectedness of the spaces Xα1 × . . . × Xαk
, homeomorphic to XF , is proved by induction on the number of factors. The proof for two factors is the essential one.
Let A and B be connected spaces. Fix a point (a, b) ∈ A × B. Then for any
x ∈ A the set Tx = ({x} × B) ∪ (A × {b}) is connected, and A × B =
S
x∈A Tx is connected (by
Proposition 43.4).
43.6. Definition. A nonempty connected set Z in a space X is called a maximal connected set or a connected component of the space X, if every connected set Z1, satisfying the condition Z ⊂ Z1 ⊂ X, coincides with Z.
Any connected component of a space X is closed in X (Proposition 43.4).
Every space X is a disjoint union of its connected components (Proposition 43.4). Hence every point x of a space X lies in a unique connected component,
which is denoted by Cx and called the connected component of the point x.
43.7. Examples. 1. In a connected space there is one connected component.
2. In the space Q the connected components are one-point sets.
3. In the product Q × R ⊂ R
2 the connected components are the vertical lines,
given by the equations x = r, where r is a rational number.
4. A space X is called locally connected, if every neighborhood of an arbitrary
point x ∈ X contains a connected neighborhood. The connected components of a locally connected space are clopen.
43.8. Proposition. The connected component of the point x = (xα) of the product X =
Q
α∈A Xα
coincides with the product Q
α∈A Cα of the connected components Cα of the points xα in the factors Xα,
α ∈ A.
72
Proof. Let Cx be the connected component of the point x = (xα). From item 43.5.2 it follows
that prα(Cx) is a connected subset of Xα and xα ∈ prα(Cx) ⊂ Cα. Hence Cx ⊂
Q
Q α∈A Cα. But
α∈A Cα is connected (item 43.5.3), and hence Cx =
Q
α∈A Cα.
43.9. Definition. The intersection of all clopen subsets of a space X
containing the point x ∈ X, is called the quasicomponent of the point x and is denoted by Qx.
43.10. Proposition. For any point x ∈ X its connected component Cx is contained
in the quasicomponent Qx.
Proof. Let Uα be any clopen set containing the point x. From Lemma 37.3 we obtain that Cx ⊂ Uα. Then Cx ⊂
T
x∈Uα
Uα = Qx.
43.11. Example of a space in which the quasicomponent does not coincide with the connected
component.
Consider the plane set X, which is the union of two vertical lines α and
β, given respectively by the equations x = −1, x = 1, and the contours of rectangles qn, two
sides of which have the equations x = −1+ 1
n
, x = 1−
1
n
, and the other two lie on the horizontal
lines y = n, y = −n, n ≥ 2.
Let us prove that the union of the two lines α and β is a quasicomponent in the space X.
Indeed, any clopen set U intersecting the line α contains
the whole of this line and the contours of all rectangles qn, starting from some n0, and, hence,
being closed, contains the line β. Thus the quasicomponent of an arbitrary point
a ∈ α contains the set α ∪ β. But Qa is the intersection of all clopen sets containing the point a. Hence this quasicomponent cannot intersect any of the
contours qn, i.e. Qa = α ∪ β. But this set is disconnected and, hence, is not a component.
43.12. Theorem. In a compact Hausdorff space X the connected component Cx
of any point x ∈ X coincides with its quasicomponent Qx.
Proof. By Proposition 43.10 it suffices to check that
Cx ⊃ Qx =
\
α∈A
Uα, (43.1)
and for this it suffices to show that the set Qx is connected. If this is not so, then Qx is represented
as a disjoint union of nonempty sets F1 and F2 closed in Qx (and hence in X). By virtue of the
normality of the compact space X, there exist disjoint neighborhoods OF1 and
OF2, whose union OQx = OF1 ∪ OF2 is a neighborhood of Qx.
Let us show that it contains a clopen neighborhood U. The family of open sets
v = {X \ Uα : α ∈ A}
covers the compact subspace X \ OQx. From v one can select a finite subfamily
{X\Uαi
: i = 1, . . . , n}, covering X\OQx. Then U =
Tn
i=1 Uαi is the required clopen
neighborhood.
Set Wi = U ∩ OFi
, i = 1, 2. From the condition U ⊂ OF1 ∪ OF2 it follows that the sets W1 and
W2 are clopen in X. The point x lies in one of these sets, say, in W1. Then the
quasicomponent Qx also lies in W1. Hence Qx∩W2 = ∅ and therefore Qx∩F2 = ∅, since
F2 ⊂ U ∩ OF2 = W2. The contradiction obtained proves the connectedness of Qx.
Assignment N 13
1. Prove that if f ∈ C(X × Y, Z), then F : X → C(Y, Z) (see item 42.1 of Lecture 13)
is continuous, where the set of maps C(Y, Z) has the topology of pointwise convergence. Give
an example where the map f ∈ C(X ×Y, Z), defined by a continuous map F : X →
C(Y, Z), where the set of maps C(Y, Z) has the topology of pointwise convergence, is not
continuous.
2. Prove that if a map F : X → C(Y, Z), where the set of maps C(Y, Z) has
the topology of uniform convergence, is continuous, then the map f ∈ C(X × Y, Z) (see item
42.1 of Lecture 13) is continuous. Give an example where the map F : X → C(Y, Z), defined by a continuous map f ∈ C(X × Y, Z), where the set of maps C(Y, Z) has the
topology of uniform convergence, is not continuous.
3. Prove that the map
73
◦ : C(X, Y ) × C(Y, Z) → C(X, Z), ◦(f, g) = g ◦ f
of spaces of maps with the compact-open topologies, is continuous.
4. Prove that the segment [0, 1] is connected.
5. Let T1 ≤ T2 be topologies on X. What can be said about the connectedness of X in one topology,
if X is connected in the other topology?
6. Prove that the union of a family of pairwise intersecting connected subsets is connected.
7. Let A ∩ B and A ∪ B be connected sets. Is it true that A and B are connected sets? What
if, in addition, both sets are open (closed)?
8. Is it true that the intersection of connected sets is connected? Will a countable intersection of connected
sets An such that A1 ⊃ A2 ⊃ . . . be connected?
9. Prove that a countable intersection of connected compact subsets of a Hausdorff space An such that A1 ⊃ A2 ⊃ . . . is connected.
10. Will the interior of a connected set be connected? Will the boundary of a connected set be
connected? Will a set be connected if its boundary is connected?
11. Let A ⊂ X. Prove that if C is a connected subspace of X, intersecting both A
and X \ A, then C ∩ BdA 6= ∅.
12. Let A and B be proper subsets of connected spaces X and Y respectively (i.e.
A 6= X, B 6= Y ). Prove that (X × Y ) \ (A × B) is connected.
13. Will the preimage under a continuous map of a connected set be connected, if the preimage of every point is connected?
14. Prove that a countable normal space is disconnected. Estimate from below the cardinality of an infinite normal connected space. Does there exist a countable Hausdorff connected space?
15. Which of the following spaces
N × [0, 1), [0, 1) × N, [0, 1) × [0, 1], [0, 1] × [0, 1)
with topologies generated by the lexicographic order on the products, are connected?
16. Prove the connectedness of the spaces R
n, the spheres S
n, and the closed balls Bn, n ∈ N.
17. Prove that R and R
n, n > 1, are not homeomorphic.
Prove that [0, 1], [0, 1) and (0, 1) are pairwise non-homeomorphic.
18. A space X is called locally connected, if every neighborhood of an arbitrary
point x ∈ X contains a connected neighborhood. Prove that the connected components of a locally
connected space are clopen.
19. Find the connected components of the following subspaces of real matrices:
(a) GL(n, R) = {A ∈ Mat(n × n, R) : det A 6= 0};
(b) O(n, R) = {A ∈ Mat(n × n, R) : AAT = E};
(c) Symm(n, R) = {A ∈ Mat(n × n, R) : AT = A}?
Additional problems of Assignment N 13
20. Prove that the map
Λ : C(X × Y, Z) → C
X, C(Y, Z)
,
of spaces of maps with the compact-open topologies, defined by the condition Λ(f) = F
(see item 42.1 of Lecture 13), is continuous for any X, Y, Z. If, moreover, Y is Hausdorff and locally
compact, then this map is a homeomorphism.
21. Prove the connectedness of a linearly ordered space X such that:
1) any bounded subset of X has a least upper bound in X;
2) if x < y, then there exists z ∈ X, x < z < y.
22. Prove that there is no continuous bijection of the line onto the square.
74
23. Prove that R
N with the topology of uniform convergence is disconnected?
24. Is the statement true that a function on a segment is continuous if and only
if the image of every segment is a segment?
25. Prove that if A is an open connected bounded set in the plane, then
there exists a unique line, parallel to a fixed line l, which divides A into
two sets of equal area.
Prove that if A and B are open connected bounded sets in the plane, then
there exists a line which divides each of the sets A and B into two sets of equal area.
Prove that if A is an open connected bounded set in the plane, then there exist two perpendicular lines which divide A into four sets of equal area.
26. Prove that for any continuous function f : S
1 → R there exists a point x such
that f(x) = f(−x).
27. Prove that the space of homeomorphisms of the segment [0, 1] with the compact-open topology has two connected components.
Comments