§ 18. Final topology with respect to a family of mappings.
18.1. Definition. Let fα : Xα → Y, α ∈ A, — be a family of mappings of spaces Xα
into a set Y . Then it is easy to check that the family of sets
U : f
−1
α (U) open in Xα, for any α ∈ A
is a topology T on Y . We call T the final topology with respect to the family of mappings {fα : α ∈ A}.
With respect to this topology all mappings fα are continuous. Moreover, T is the largest
(strongest) topology on Y with this property.
18.2. Example.
Let Tα, α ∈ A, — be a family of topologies on a set X, idα : Xα → X, α ∈ A, — the family
of identity mappings of the spaces Xα onto the set X. Then on X the exact
lower bound of the family of topologies A is defined as the final topology with respect to the family {idα :
α ∈ A}.
18.3. Proposition. Let fα : Xα → Y, α ∈ A, — be a family of mappings of spaces
Xα into a space (Y, T ). Then the following conditions are equivalent:
1) T — is the final topology on Y with respect to the family of mappings fα, α ∈ A,
2) for any space Z a mapping f : Y → Z is continuous if and only
if the mappings f ◦ fα : Xα → Z, α ∈ A, are continuous.
Proof. 1) =⇒ 2). Necessity in condition 2) follows from the continuity of the mappings fα, α ∈ A, and their composition with f. Sufficiency. Let the set O be open in Z. Then
the set
(f ◦ fα)
−1O = f
−1
α (f
−1
(O))
is open in Xα for any α ∈ A. By Definition 18.1 the set f
−1
(O) is open in Y . Hence
the mapping f is continuous.
2) =⇒ 1). Since the identity mapping id of the space (Y, T ) into (Y, T ) is a homeomorphism, by condition 2) the compositions id◦ fα are continuous, α ∈ A. Hence the mappings fα, α ∈ A,
are also continuous, since for any open subset O ⊂ Y we have
f
−1
α (O) = (id ◦ fα)
−1
(id(O)),
and the set (id ◦ fα)
−1
(id(O)) is open in Xα, α ∈ A.
Let T
0 — be the final topology on Y with respect to the family of mappings fα, α ∈ A. Then
T
0 ≥ T , and for any α ∈ A the composition of fα : Xα → Y and the identity mapping of (Y, T )
into (Y, T
0
) is continuous (T
0 — is the final topology on Y with respect to the family of mappings
fα, α ∈ A, and on Xα the mappings fα and id ◦ fα into the set Y coincide). By condition 2) the identity mapping is continuous, i.e. any set open in (Y, T
0
) is open in (Y, T ).
Hence T ≥ T 0
. Thus, T = T
0
.
§ 19. Sum of spaces and sums of mappings.
19.1. Definition. Let Xα, α ∈ A, — be a disjoint family of topological spaces. Consider the set X =
S
{Xα : α ∈ A} and the family of identity embeddings
iα : Xα → X, α ∈ A. The set X with the final topology with respect to the family of embeddings
iα, α ∈ A is called the sum of the spaces Xα, α ∈ A, and is denoted by ⊕
α∈A
Xα or by
X1 ⊕ X2 ⊕ . . . ⊕ Xk, if A = {1, 2, . . . , k} is finite.
A set O ⊂ ⊕{Xα : α ∈ A} is open if and only if O ∩ Xα is open in Xα
for every α ∈ A. A set F ⊂ ⊕{Xα : α ∈ A} is closed if and only if F ∩Xα
is closed in Xα for every α ∈ A. All the sets Xα are open-closed in ⊕{Xα : α ∈ A}.
19.2. Corollary. Let X — be the sum of the spaces Xα, α ∈ A. A mapping f : X → Z
is continuous if and only if all compositions f ◦ iα : Xα → Z are continuous.
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19.3. Direct sum of mappings. Let sums X = ⊕{Xα : α ∈ A} and Y = ⊕{Yα :
α ∈ A} of topological spaces Xα and Yα be given, and mappings fα : Xα → Yα. Then there exists
a unique mapping f : X → Y , satisfying the relations
f ◦ iXα = iYα ◦ fα, α ∈ A,
where iXα
: Xα → X, iYα
: Yα → Y — are the embedding mappings. The mapping f is called the direct
sum of the mappings fα and is denoted by ⊕{fα : α ∈ A}.
19.4. Proposition. The direct sum f of the mappings fα : Xα → Yα, α ∈ A, is continuous
if and only if every mapping fα is continuous.
Proof. Sufficiency follows from the continuity of the compositions iYα
◦ fα, α ∈ A, and
Proposition 18.3. Necessity. Since f ◦iXα is continuous, the compositions iYα
◦fα : Xα → Y ,
α ∈ A, are continuous. If Oα is open in Yα, then iYα
(Oα) is open in Y and i
−1
Xα
(f
−1
(iYα
(Oα))) =
f
−1
α (i
−1
Yα
(iYα
(Oα)) = f
−1
α (Oα) is open in Xα. Hence, the mappings fα, α ∈ A, are continuous.
19.5. Sum of mappings. For a family of spaces Xα, α ∈ A, and a family of mappings fα : Xα → Y, α ∈ A, into a space Y there exists a unique mapping f : ⊕{Xα :
α ∈ A} → Y , satisfying the relations
f ◦ iXα = fα, α ∈ A. (19.1)
The mapping f is called the sum of the mappings fα and is denoted by P{fα : α ∈ A}.
19.6. Proposition. The sum f of the mappings fα : Xα → Y , α ∈ A, is continuous if and
only if every mapping fα is continuous.
Proof. If the mapping f is continuous, then according to (19.1) each mapping
fα is continuous as a composition of the continuous mappings iXα and f. Sufficiency follows from
Proposition 18.3.
§ 20. Quotient spaces and quotient mappings.
20.1. Definition. Let f : X → Y — be a mapping of a space X onto a set Y .
The final topology
T =
U ⊂ Y : f
−1
(U) is open in X
with respect to the mapping f is called the quotient topology (or factor topology), the space (Y, T ) — the quotient space of the space X, and the mapping f — the quotient
mapping.
It is obvious that the quotient topology is the strongest among all topologies on X for which the mapping f : X → Y is continuous.
20.2. Examples. Quotient mappings arise naturally from partitions of a space X into disjoint sets.
1) Take the partition R of the real line R into two sets P and Q of irrational and
rational numbers respectively. Then the quotient set R/R consists of two points a and b.
As for the quotient topology, this is the topology of the “glued-together two-point set”, i.e. the weakest topology
{∅, R/R} on the quotient set.
2) Partition the square I
2 ⊂ R
2
with vertices (0, 0),(0, 1),(1, 0) and (1, 1) into vertical segments.
The resulting quotient space is homeomorphic to the segment [0, 1] ⊂ R.
3) The real projective space RP
n, n ∈ {0}∪N, — is the quotient space of the sphere
S
n by its partition into pairs of diametrically opposite points ((x1, . . . , xn+1) ∼ (−x1, . . . , −xn+1)).
20.3. Contracting a space.
Factorization of a space X by the partition into a set A ⊂ X and one-point sets
of the complement X \A is called contracting the set A to a point. Notation for the quotient space: X/A.
20.4. Examples.
1) The partition R of the segment [0, 1] into one-point sets of the interval (0, 1) and the two-point
set {0, 1} (contracting the endpoints of the segment to a point). Then the quotient space [0, 1]/{0, 1}
is homeomorphic to the circle S
1
.
2) The partition R of the closed disk
B2 =
(x1, x2) ∈ R
2
: x
2
1 + x
2
2 ≤ 1
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consists of one-point sets of the disk D2 ⊂ B2 and the boundary circle S
1 = B2 \ D2
(contracting the circle to a point). Then the quotient space B2/S1
is homeomorphic to the sphere S
2
.
20.5. Gluing spaces together. Let X, Y — be topological spaces, A — a subset of the space X, f : A → Y — a continuous mapping. Take the sum X ⊕ Y and
consider its partition R into one-point sets iX(X \ A) and iY (Y \ f(A)), and the sets
iY (x) ∪ iX(f
−1
(x)), x ∈ f(A). The quotient space X ⊕ Y /R is denoted by X ∪f Y , and the described procedure of its construction is called gluing X to Y by means of the mapping
f.
20.6. Example.
The wedge (bouquet) of spaces X and Y with marked points x0 ∈ X and y0 ∈ Y is the gluing of
X to Y by means of the mapping f : {x0} → {y0}. Notation: X ∨ Y .
Assignment N 5
1. Prove that the quotient space [0, 1]/{0, 1} (contracting the endpoints of the segment to a point) is homeomorphic to the circle S
1
.
Consider on the line R the equivalence relation R: x ∼ y ⇐⇒ x − y ∈ Z. Prove
that the quotient space of the line R by its partition into classes of the equivalence relation R
is homeomorphic to the circle S
1
.
2. Prove that the quotient space B2/S1
of the closed disk B2 in the plane by the boundary sphere S
1
(contracting the circle to a point) is homeomorphic to the sphere S
2
.
Prove that R
2/B2
is homeomorphic to R
2
.
3. Prove that the quotient space of the square I × I = [0, 1] × [0, 1] by its partition into
one-point subsets of the square without two sides (0, 1) × I and two-point subsets
{(0, t),(1, t)}, t ∈ I, is homeomorphic to the cylinder S
1 × I.
4. Prove that the quotient space of
(a) the cylinder S
1 × I by its partition into one-point subsets
S
1 × (0, 1) and two-point subsets {(t, 0),(t, 1)}, t ∈ S
1
,
(b) the square I × I by its partition into one-point subsets of the square without sides
(0, 1)×(0, 1), two-point subsets {(0, t),(1, t)}, {(t, 0),(t, 1)}, t ∈ (0, 1), and the subset {(0, 0),(1, 0),(0, 1),(1, 1)}
is homeomorphic to the torus S
1 × S
1
.
5. Prove that the quotient space of the square I ×I by its partition into one-point subsets of the square without sides (0, 1)×(0, 1) and two-point subsets {(0, t),(t, 0)}, {(t, 1),(1, t)},
t ∈ I, is homeomorphic to the sphere S
2
.
6. The quotient space of the square I ×I by its partition into one-point subsets of the square without sides (0, 1) × [0, 1] and two-point subsets {(0, t),(1, 1 − t)}, t ∈ I, is called
the Möbius band.
The quotient space of the square I×I by its partition into one-point subsets of the square
without sides (0, 1) × (0, 1) and two-point subsets {(0, t),(1, t)}, {(t, 0),(1 − t, 1)}, t ∈ (0, 1), and
the subset {(0, 0),(1, 0),(0, 1),(1, 1)} is called the Klein bottle.
Represent the Klein bottle as the result
(a) of factorizing the cylinder S
1 × I,
(b) of factorizing the Möbius band,
(c) of gluing two copies of the Möbius band along their boundaries by means of the identity mapping,
(d) of gluing two copies of the cylinder along their boundaries.
7. Prove that the quotient space of the square I × I by its partition into one-point
subsets of the square without sides (0, 1) × (0, 1) and two-point subsets {(0, t),(1, 1 −
t)}, {(t, 0),(1 − t, 1)}, t ∈ I, is homeomorphic to the projective plane RP
2
.
8. Obtain the projective plane RP
2
(a) as the result of gluing the closed disk B2 and the Möbius band along their boundary,
(b) as the result of factorizing the Möbius band.
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9. Prove that metrizable spaces satisfy the first countability axiom, the Sorgenfrey line satisfies the first countability axiom, and the line with the Zariski topology does not
satisfy the first countability axiom.
10. Prove that if a space satisfies the second countability axiom, then it satisfies the first countability axiom.
11. Give an example of a metrizable space that does not satisfy the second countability
axiom.
12. Prove that in every subset of R (with the standard topology) there is a countable
everywhere dense subset.
13. Are the first (second) countability axiom and separability of a space preserved in the image under continuous mappings?
14. Prove that a subset Z of a topological space X is everywhere dense in X if and
only if Z ∩ O 6= ∅ for any nonempty open subset O
of the space X.
15. Is the intersection of two everywhere dense subsets everywhere dense? What if, additionally, one of the subsets is open?
Is the intersection of a countable number of open everywhere dense subsets everywhere dense?
16. A subset A of a space X is nowhere dense if the set X \ClA is everywhere dense.
Prove that the Cantor set is nowhere dense in R.
17. Prove that if A is a nowhere dense subset, then ClA is also a nowhere dense
subset.
18. Prove that the boundary of a closed (open) set is nowhere dense. Give
an example of a space and its subset with an everywhere dense boundary.
19. Check the truth of the statements:
(a) the continuous image of an everywhere dense set is everywhere dense in the image;
(b) the continuous image of a nowhere dense set is nowhere dense in the image.
Additional problems for Assignment N 5
20. Let f : X → Y be a continuous mapping. Prove that if there exists a continuous
mapping g : Y → X such that f ◦ g = idY , then f — is a quotient mapping.
Let A ⊂ X. A continuous mapping f : X → A, for which f(a) = a, a ∈ A, is called
a retraction of X onto A, and the subset A is called a retract of X.
Prove that a continuous mapping f : X → X is a retraction onto its image f(X)
if and only if f ◦ f = f.
Prove that a retraction is a quotient mapping.
21. Consider on the line R the equivalence relation R: x ∼ y ⇐⇒ x − y ∈ Q. Find
the quotient space R/R.
22. Prove that any uncountable closed subset of the line R has the cardinality of the continuum.
Prove that any nonempty closed subset of the line R without isolated points
has the cardinality of the continuum.
23. Prove that if a space satisfies the second countability axiom, then from any
base of it one can choose a countable subfamily that is also a base.
24. Does the quotient space of a space satisfying the first (second) countability
axiom satisfy the first (second) countability axiom?
25. Prove that the Tychonoff product of continuum many separable spaces is separable.
26. Does there exist a countable space that does not satisfy the first countability axiom?
27. Find (describe) all topologies on a set X for which the one-point set {x}, where x ∈ X, is everywhere dense.
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