Lecture
§ 44. Hereditary disconnectedness. Zero-dimensionality.
44.1 Definition. A space X is called totally disconnected or hereditarily disconnected, if every subspace of it containing more than one point is disconnected.
Every totally disconnected space is a T1-space.
A space is totally disconnected if and only if its connected components are singletons.
44.2. Definition. A space X is called inductively zero-dimensional, if it has
a base consisting of open-closed sets.
An inductively zero-dimensional T0-space X is a Tychonoff space, and is also
called a zero-dimensional space.
44.3. Properties of disconnectedness. A subspace of an inductively zero-dimensional (respectively
totally disconnected) space is inductively zero-dimensional (respectively totally disconnected).
A zero-dimensional space is totally disconnected.
The Tychonoff product of inductively zero-dimensional (respectively zero-dimensional, totally disconnected) spaces is inductively zero-dimensional (respectively zero-dimensional, totally disconnected) (Proposition 43.8 of Lecture 13).
44.4. Examples. 1. The Cantor set is zero-dimensional, and hence also totally disconnected. Indeed, the intersection of intervals of the line, whose endpoints belong to the complement of the Cantor
set, with the Cantor set, is an open-closed base.
2. The subspaces Q ⊂ R (respectively P ⊂ R) of rational (respectively irrational) numbers of the real line R are zero-dimensional.
3. A totally disconnected non-zero-dimensional space.
Consider a rectangular coordinate system Oxy on the plane R
2
. On the segment [0, 1] of the
Ox axis consider the Cantor set C. Join each point x
of the set C by a straight line segment to the point a = (1/2, 1/2) of the plane R
2 and denote this segment by [a, x]. If
the point x ∈ C is an endpoint of an interval complementary to C, then on the segment [a, x] we take all the points whose
second coordinate is rational, otherwise on the segment [a, x] we take all the points whose
second coordinate is irrational. All the chosen points make up the “Knaster–Kuratowski fan,” which we denote by K.
Problem. Prove that the subspace K \ {a} is totally disconnected, but not zero-dimensional
(Note that the space K is connected.)
44.5. Proposition. Zero-dimensionality and total disconnectedness of Hausdorff compact spaces coincide.
Proof. By virtue of the properties of disconnectedness (item 44.3) it is necessary to show that a totally
disconnected compact space X is zero-dimensional. Let O be a neighborhood of a point x ∈ X. By
Theorem 43.12 of Lecture 13 its quasicomponent Qx =
T
α∈A Uα = {x}. From the family {X\Uα : α ∈ A}
of open sets covering the compact subset X \ O, one can choose a finite
subfamily {X \ Uα1
, . . . , X \ Uαk
}, covering X \ O. Then Tk
i=1 Uαi
is an open-closed
neighborhood of the point x, and Tk
i=1 Uαi ⊂ O. Thus it is established that X has a base of open-closed sets.
§ 45. Path-connectedness.
45.1. Definition. A continuous mapping ϕ : I = [0, 1] → X is called a path in
the space X. The points ϕ(0) and ϕ(1) are called respectively the beginning and end of the path ϕ. If
the beginning and end of the path ϕ coincide, then the path ϕ is called a loop.
45.2. Definition. A space X is called path-connected, if any two of its points
can be joined by a path, i.e., for any two points x, y ∈ X there exists a path ϕ such that
ϕ(0) = x, ϕ(1) = y.
Every path-connected space is connected.
45.3. Example. The compact set “sin 1
x
” from example 31.2.3 of Lecture 9 is connected (the connected subset
X2 is dense everywhere), but not path-connected. The points (0, 0) ∈ X1 and (x
0
, y0
) ∈ X2 cannot be joined by a path.
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Indeed, suppose there exists a path
ϕ : I → X, ϕ(t) = (x(t), y(t)), ϕ(0) = (0, 0), ϕ(1) = (x
0
, y0
).
The set ϕ
−1
(X1) is closed in [0, 1] and let τ = sup{t : t ∈ ϕ
−1
(X1)}. Then τ < 1, ϕ(τ ) ∈ X1,
ϕ((τ, 1]) ⊂ X2.
For any n ∈ N there exists τ < tn < min{1, τ +
1
n
} such that y(tn) = (−1)n. Indeed,
x(min{1, τ +
1
n
}) > 0. Hence there exists a, 0 < a < x(min{1, τ +
1
n
}) such that sin( 1
a
) = (−1)n.
The image of the segment [τ, min{1, τ +
1
n
}] contains the segment [0, x(min{1, τ +
1
n
})]. Hence there exists a point
tn ∈ [τ, min{1, τ +
1
n
}] such that x(tn) = a. Since (x(tn), y(tn)) ∈ X, we have y(tn) = (−1)n.
We have limn→∞ tn = τ , but limn→∞ ϕ(tn) = limn→∞(x(tn), y(tn)) is not defined. Hence the mapping ϕ is discontinuous at the point τ (Theorem 14.7 of Lecture 3).
45.4. Definition. Let ϕ : I → X and ψ : I → X be paths such that ϕ(1) = ψ(0). The product of the paths ϕ and ψ is the path ϕψ : I → X,
ϕψ(t) =
ϕ(2t) for 0 ≤ t ≤
1
2
,
ψ(2t − 1) for 1
2 ≤ t ≤ 1.
The reverse path to ϕ : I → X is the path ϕ
−1
: I → X,
ϕ
−1
(t) = ϕ(1 − t).
45.5. Proposition. Let Zα be path-connected subsets of a space X, α ∈ A.
If Z0 =
T
α∈A Zα 6= ∅, then the set Z =
S
α∈A Zα is path-connected.
Proof. Let x, y ∈ Z be such that x ∈ Zα1
, y ∈ Zα2
, and x0 ∈ Z0. Let us find a path with
beginning at x and end at y.
There exist paths ϕ : I → Zα1
, ϕ(0) = x0, ϕ(1) = x, and ψ : I → Zα2
, ψ(0) = x0, ψ(1) = y. The path
ϕ
−1ψ : I → Z is the required one.
45.6. Properties of path-connectedness.
1. The continuous image of a path-connected space is path-connected.
2. The Tychonoff product of path-connected spaces is a path-connected space.
Let X =
Q
α∈A Xα and Xα be a path-connected space, α ∈ A. For x = (xα), y = (yα) ∈ X
there exist paths ϕα : I → Xα, ϕα(0) = xα, ϕα(1) = yα, α ∈ A. Then the path ϕ = ∆α∈Aϕα is the
required one.
45.7. Definition. A maximal (with respect to inclusion) path-connected subset of a space X is called a component of path-connectedness of the space X.
Unlike connected components, components of path-connectedness need not be closed. Thus, the compact set “sin 1
x
” has two components of path-connectedness: the closed X1 and
the non-closed X2.
The space X is the disjoint sum of its components of path-connectedness (see
Proposition 45.5). Every point x ∈ X is contained in a unique component of path-connectedness of the space X.
§ 46. Homotopy. Homotopies relative to a subset.
46.1. Definition of homotopy. Continuous mappings f : X → Y and g : X → Y are called homotopic (notation f ∼h g), if there exists a continuous mapping
Φ : X × I → Y , such that Φ(x, 0) = f(x), Φ(x, 1) = g(x) for every x ∈ X. Every such mapping
Φ is called a homotopy connecting f with g.
A mapping homotopic to a constant mapping is called null-homotopic.
Sometimes the mapping Φ is replaced by the family of mappings
ft : X → Y, t ∈ I, ft(x) = Φ(x, t).
The family of mappings {ft : X → Y : t ∈ T} can be regarded as a mapping
F : I → C(X, Y ).
From Theorem 42.2 of Lecture 13 it follows.
46.2. Proposition. Let the space C(X, Y ) have the compact-open topology.
If the mapping Φ : X × I → Y is continuous, then the mapping F : I →
C(X, Y ) is also continuous.
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If the mapping F : I → C(X, Y ) is continuous and the space X is locally compact, then
the mapping
Φ : X × I → Y, Φ(x, t) = ft(x)
is continuous.
46.3. Theorem. The homotopy relation is an equivalence relation on the set C(X, Y ) of all continuous mappings from X to Y .
Proof. Reflexivity. Every mapping f : X → Y is homotopic to itself.
Indeed, it suffices to consider the constant homotopy of the mapping f: Φ : X ×I → Y ,
where Φ(x, t) = f(x) for any x ∈ X, t ∈ I. Its continuity follows from the formula Φ = f ◦ prX,
where prX is the projection of the product onto the factor X.
Symmetry. Let the mapping Φ : X ×I → Y be a homotopy connecting the mappings
f and g. Consider the mapping Γ : X × I → Y , defined by the formula Γ(x, t) = Φ(x, 1 − t). This
mapping is obviously continuous and is a homotopy connecting the mappings g and f.
Consequently, the relation ∼h is symmetric.
Transitivity. Let mappings fi
: X → Y, i = 1, 2, 3 be given. Suppose f1 ∼h f2, and
f2 ∼h f3. We must show that f1 ∼h f3. Let the homotopy Φ1 connect f1 with f2, and the homotopy Φ2
connect f2 with f3. Define the homotopy Φ : X × I → Y as follows:
Φ(x, t) =
F1(x, 2t) for 0 ≤ t ≤
1
2
;
F2(x, 2t − 1) for 1
2 ≤ t ≤ 1.
We have Φ
x, 1
2
= Φ1(x, 1) = f2(x). On the other hand Φ
x, 1
2
= Φ2(x, 0) = f2(x). Hence, the mapping Φ is well defined. It is continuous, being continuous on the closed summands
X ×
0,
1
2
and X ×
1
2
, 1
. Finally, Φ(x, 0) = Φ1(x, 0) = f1(x, 0), Φ(x, 1) = Φ2(x, 1) = f3(x). Thus,
the homotopy Φ connects f1 with f3.
46.4. Definition. The equivalence classes of homotopic mappings are called homotopy classes. The homotopy class of a mapping f : X → Y will be denoted by
[f]. The set of homotopy classes of mappings X → Y is denoted by π(X, Y ).
46.5. Examples. 1. π(X, I) consists of a single point. Indeed, let f0, f1 : X → I be two
continuous mappings. Consider the homotopy Φ : X × I → I, defined as
follows:
Φ(x, t) = (1 − t)f0(x) + tf1(x). (46.1)
It is clear that the mapping from (46.1) connects f0 with f1.
The continuity of the mapping Φ(x, t) = (1 − t)f0(x) + tf1(x) = f0(x) + t(f1(x) − f0(x)) can
be proved by establishing the continuity of the mappings H(x, t) = f0(x) : X ×I → I and G(x, t) = t(f1(x)−
f0(x)) : X × I → I (as a sum of continuous functions). It remains to note that H = f0 ◦ prX
(and it is continuous as a composition of continuous mappings), G(x, t) = prI (x, t) · ((f1 − f0) ◦
prX)(x, t) (and it is continuous as a product of continuous functions), where prI , prX are the projections
of the product X × I onto the factors I and X respectively.
2. The arguments of the previous item show that π(X, V ) is a single point for any convex subset R
n.
3. If X is a one-point space, then the set π(X, Y ) is the set of components of path-connectedness of the space Y .
4. If the space X is locally compact, then π(X, Y ) is the components of path-connectedness
of the space C(X, Y ) in the compact-open topology.
46.6. Lemma. If h : A → X, f, f0
: X → Y, g : Y → B are continuous mappings, and
F : X ×I → Y is a homotopy connecting f and f
0
, then g ◦F ◦ (h×id) is a homotopy connecting
g ◦ f ◦ h and g ◦ f
0 ◦ h : A → B, where id is the identity mapping of the segment I.
Proof. A trivial verification of the conditions of homotopy of mappings.
46.7. Homotopies relative to a subset. Let A be a subset of a space X. A homotopy F :
X × I → Y is called relative to A, or an A-homotopy, if F(x, t) = F(x, 0) for x ∈ A,
t ∈ I. Two mappings that can be connected by an A-homotopy are called A-homotopic.
Like ordinary homotopy, A-homotopy is an equivalence relation. As a
rule, we will deal only with a one-point and a two-point set A.
Problem Set N 14
1. Prove that a totally disconnected space is a T1-space.
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2. Prove that a zero-dimensional space is totally disconnected.
3. Prove that the subspaces Q ⊂ R (respectively P ⊂ R) of rational (respectively irrational) numbers of the real line R are zero-dimensional.
4. Prove that the Sorgenfrey line is zero-dimensional.
4. Prove that the Tychonoff product of zero-dimensional (respectively totally disconnected)
spaces is zero-dimensional (respectively totally disconnected).
5. Let A∩B and A∪B be path-connected sets. Is it true that A and B are path-connected
sets? And if, additionally, both sets are open (closed)?
6. Is the interior of a path-connected set connected? Is the boundary of a path-
connected set connected? Is a set path-connected if its boundary is path-
connected?
7. 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 path-connected?
8. Prove the path-connectedness of the spaces R
n, spheres S
n and closed balls Bn, n ∈ N.
9. Find the components of path-connectedness 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}?
10. Prove that for subsets of the line, connectedness and path-connectedness are equivalent.
Prove that an open connected subset of R
2 is path-connected.
11. Let A be a countable subset of R
2
. Prove that R
2 \ A is path-connected.
12. Prove that π(X, V ) is a single point for any convex subset V in R
n.
13. Establish the homotopy of any continuous non-surjective mappings f, g : X →
S
n, n ∈ N.
14. Prove that if X is a one-point space, then the set π(X, Y ) is the set of
components of path-connectedness of the space Y .
15. When are two constant mappings homotopic?
16. Let X be a path-connected space. Prove that π(I, X) is a single point.
17. Prove that if the space X is Hausdorff and locally compact, then π(X, Y ) is
the components of path-connectedness of the space C(X, Y ) in the compact-open topology.
18. Prove that if the mappings f, f0
: X → Y are homotopic and the mappings g, g0
: Y → Z
are homotopic, then the mappings g ◦ f and g
0 ◦ f
0
are homotopic.
19. Prove that continuous mappings f, g : X → Y ×Z are homotopic if and only if the pairs of compositions pY ◦ f, pY ◦ g and pZ ◦ f, pZ ◦ g are homotopic, where pY and pZ are the projections
in the product onto the corresponding factors.
Additional Problems for Problem Set N 14
20. Prove that the space Knaster–Kuratowski fan K (Example 37.4 of Lecture 14)
is connected, and the subspace K \ {a} is totally disconnected, but not zero-dimensional.
21. Prove that the set of points of Hilbert space `
2 all of whose coordinates are
rational is totally disconnected, but not zero-dimensional.
22. Prove that the quotient space of a Hausdorff compact space by its partition into connected components is a zero-dimensional Hausdorff compact space.
23. Let X1 and X2 be the closed and open components of path-connectedness of the compact set X
from Example 31.2.3 of Lecture 9. Prove that if under a continuous mapping f : X → X
there exists a point x ∈ X2, for which f(x) ∈ X1, then also f(X) ⊂ X1.
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24. Will any two continuous mappings of an arbitrary space
X into a path-connected space Y be homotopic?
25. Give an example of spaces X and Y, a subset A of the space X and continuous
mappings f, g : X → Y such that f|A = g|A, f and g are homotopic, but not A-homotopic.
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