Categories and
Topology
ZWEISTEIN
Desu-Cartes
1
Contents
1 Introduction 3
1.1 What is the purpose of this lecture? . . . . . . . . . . . . . . . . . . . . . 3
1.2 What is topology? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 A few definitions 4
2.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Topology, Continuity, Convergence . . . . . . . . . . . . . . . . . 4
2.1.2 Connectedness and Path-Connectedness . . . . . . . . . . . . . . 7
2.1.3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Categories and morphisms . . . . . . . . . . . . . . . . . . . . . . 9
2.2.2 Functors and diagrams . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Topological constructions 12
3.1 Final and Initial topology . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2.1 First attempt: The Box Topology . . . . . . . . . . . . . . . . . . . 13
3.2.2 Second attempt: The Product Topology . . . . . . . . . . . . . . . 15
3.3 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.4 Dual of the product topology: disjoint unions . . . . . . . . . . . . . . . 16
3.5 Dual of the subspace topology: quotient spaces . . . . . . . . . . . . . . 16
3.6 Weak-
∗
topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2
1 Introduction
1.1 What is the purpose of this lecture?
I decided to make this lecture after realizing that many basic constructions in topol-
ogy can be explained via commutative diagrams. Category theory is also the natural
setting to understand the basic ideas behind algebraic topology. Indeed, homotopy
and homology groups are best understood as functors between categories. Cate-
gory theory actually originated in algebraic topology with the works of S. MacLane
and S. Eilenberg. From this point onwards, categories have become useful in many
other areas of mathematics and science, from algebraic geometry to mathematical
physics and theoretical computer science.
I do not assume anything more than good familiarity with set theory (includ-
ing disjoint unions and equivalence relations) and some basic understanding of
real analysis. Although I sometimes use examples from other areas of mathemat-
ics, they are certainly not essential and are just there to accommodate everyone’s
background. A course on point-set topology can be useful to understand the con-
structions here since they’re the same, except for the fact that the viewpoint used
here might be slightly different depending on your course. This lecture is not in-
tended to replace a course on topology, it is meant to be concise and I only develop
the notions that I need in order to construct the topologies. Many easy results are
left as exercises. You’re free to skip them if you want to. The section on the weak-*
topology is slightly different from the rest of the material and probably needs some
understanding of linear algebra or functional analysis to be fully appreciated.
1.2 What is topology?
A tenet of modern mathematics is that morphisms between mathematical struc-
tures are even more useful than the underlying structures. In set theory, we see that
bijections preserve the set structure. If A, B are two sets and we have a bijection
ϕ : A → B, then A and B are ‘essentially the same’. What this means is that you can
re-label every element of A with elements of B and not lose any information. In
other words, you do not lose any element in the process (surjection) and two dis-
tinct elements in B were distinct in A as well (injection). A bijection thus preserves
the set structure, but if our set has more structure on top of it, it is certainly not
guaranteed that our bijection preserves that structure as well. In full generality, a
map that fully preserves a mathematical structure is called an isomorphism. An iso-
morphism between vector spaces preserves the linear structure. An isomorphism
between groups preserves the group structure.
The question of topology is then, what do topological isomorphisms preserve?
What kind of structures do topological spaces hold? To put it simply, topology is con-
cerned with continuity, and topological isomorphisms (that we will henceforth call
homeomorphisms) are required to be continuous. But of course, topological spaces
are not restricted to subsets of R, which means that continuity on those spaces will
have a more general meaning than the usual ε −δ definition of continuity. Defining
a topology on a space will give a precise meaning to basic intuitions about conver-
3
gence, continuity, connectedness and compactness which form the crux of elemen-
tary real analysis. Topology can be seen as an extension of geometry and studies
both local and global properties.
2 A few definitions
2.1 Topology
2.1.1 Topology, Continuity, Convergence
After setting the stage for what we will be concerned in this lecture, we now turn to
the basic definition of a topology on a set. Throughout this section, X stands for an
arbitrary set.
DEFINITION 2.1.1. A topology on a set X is a collection O of subsets of X such that:
1. X and ∅ are both in O
2. If {A
i
}
i∈I
is a possibly infinite family of elements of O, then their union is in O .
3. If {A
i
}
i∈I
is a finite family of elements of O , then their intersection is in O .
Elements of O are called open sets. The pair (X ,O ) is called a topological space.
In other words, a topology on X is a collection of subsets of X that are stable un-
der arbitrary union, finite intersection and contain both the empty set and X itself.
Two somewhat natural topologies on any set X usually spring in mind when one
first tries to come up examples of topologies.
EXAMPLE 2.1.2. The collection of subsets of X defined by O = {X ,∅} is a topology.
We usually call it the trivial topology because it doesn’t hold much information
(there are only two open sets). Similarly, the power set of X is also a topology since
every subset of X is included. We call this topology the discrete topology. Single-
tons themselves are open in this topology, so there is an enormous amount of open
sets. Both topologies are usually not very useful, because in one case the topology is
too thin and doesn’t hold much information, in the other case the topology usually
holds too much information.
DEFINITION 2.1.3. If we have two topologies O
1
and O
2
, O
1
is said to be finer than O
2
if O
2
⊆O
1
. In this case, O
2
is said to be coarser than O
1
.
These examples, while simple, won’t interest us as much as other topologies. We
have now defined the notion of a topological space, but it is unclear why it is re-
lated to convergence and continuity. Naturally, our definitions of continuity here
will need to coincide with the definitions in R. But first, we need to find a topology
on R, because we haven’t yet shown anything about R. Unsurprisingly, the open sets
correspond exactly to the open sets we see in analysis.
DEFINITION 2.1.4. The standard topology on R denoted O
R
is precisely the set of
open sets in R. A set is open in R precisely if it is of the form
S
i=1
U
i
where U
i
are
open intervals.
4
EXERCISE 2.1.5. Show that O
R
is a topology by checking directly the axioms.
In a sense, we can see that the open intervals of R form a basis for the topology on
R. While we won’t need this construction, it is a very useful way to define a topology.
First you define a basis of sets with certain properties, and then you say the topology
generated by that basis is the set of all countable unions of those basis elements.
We now proceeds to define what we mean by converging and continuity. To this
end, we need to understand how convergence in R works. For a sequence to con-
verge in R, we require it to get closer and closer to our limit after a certain index N.
While we do not have a general definition of distance in topological spaces, we do
have a notion of ‘closeness’. X now stands for a topological space (X , O
X
).
DEFINITION 2.1.6. Let p ∈ X be a point in our topological space. We say V is a neigh-
borhood of p if there exists an open set U ⊆V such that p ∈U.
The name neighborhood is useful here in visualizing the situation. X itself is
a neighborhood of every point p ∈ X (why?). In R, we can find neighborhoods of
points rather easily. Set ε > 0. Then the set (p −ε, p +ε) is a neighborhood of p
(why?). Another way to say that a sequence (x
i
)
∞
i=1
converges to p is to say that there
exists an N ∈N such that for all n ≥N, x
n
∈(p −ε,p +ε).
EXERCISE 2.1.7. Show that this definition is equivalent to the usual definition of con-
vergence of a sequence in real analysis.
This viewpoint will be much more useful to define convergence in general topo-
logical spaces.
DEFINITION 2.1.8. Let (x
i
)
∞
i=1
be a sequence of elements in X . We say that the se-
quence converges to x if for every neighborhood V of x, there exists N ∈N such that
for all n ≥ N, x
n
∈V . We write this as lim
n→∞
x
n
=x.
To get a sense of how convergence in general topological spaces differ from con-
vergence in R, we will now test these definitions in the case of the discrete and trivial
topology.
EXAMPLE 2.1.9 (Convergence in the discrete topology). Let X be an arbitrary set and
equip X with the discrete topology. Since every subset of X is open, the singletons
themselves are open. Let p be a point in X . Since {p} is open, {p} is a neighborhood
of p. This is a very stringent condition on the convergence of our sequence, because
it means every sequence converging to p must be the constant sequence p after a
certain N . Indeed, the only way for the sequence to be in {p} for all n ≥ N is for x
n
to be equal to p.
EXAMPLE 2.1.10 (Convergence in the trivial topology). Again, let X be an arbitrary
set and equip X with the trivial topology. This time, things are even weirder. Since
the only open sets in X are X and ∅, and p ∉ ∅ for every p, then the only neighbor-
hood of p is X . Surely, since x
i
∈ X for all i , then our sequence is always contained
in our neighborhood. Thus every sequence converges in this topology.
5
Those two examples show two extremes. On one hand, every sequence con-
verges, on the other, only eventually constant sequences converge. The standard
topology on R lies in between these two topologies. We already see a link between
the number of open sets and how easily a sequence converges. The more open sets
there are, the harder it is for a sequence to converge in the topology.
We now turn to the other important definition in this chapter, that of a continu-
ous function.
DEFINITION 2.1.11. Let X , Y be two topological spaces. A function f : X → Y is said
to be continuous if for every open set V ∈O
Y
, f
−1
(
V
)
is open in X . In other words:
f
−1
(
O
Y
)
⊆O
X
It is interesting to note that we require the preimage of an open set to be open
in the topology of X , but not the image of an open set to be open in Y . If f : X → Y
is such a map, we say f is an open map. A homeomorphism is thus a bicontinuous
bijection, that is to say, both f (x) and f
−1
(
x
)
are continuous functions from X to Y .
If there is a homeomorphism between two spaces, we say they are homeomorphic,
and we write X ' Y . Two homeomorphic spaces are topologically the same, just like
two sets in bijection are set-theoretically the same. The homeomorphism preserves
the topological structure and we’re effectively just re-labeling elements and open
sets.
It is not obvious how this topological definition of continuity is related to the
usual ε−δ definition of continuity. We now prove that they’re the same for the stan-
dard topology on R. To this end, we start by defining ‘open balls’.
DEFINITION 2.1.12. Let ε >0 be a real number. The set B
ε
(x) :={y ∈R ||x −y|<ε} is
called the open ball of radius ε and center x.
We can now redefine open sets in R in terms of open balls. A subset A of R is
open if for every x ∈ A, there exists a δ > 0 in R such that B
δ
(x) ⊆ A. We leave the
proof of the equivalence as an exercise. You are free to give a proof by authority say
it is trivial if you wish to do so. We now turn to our first theorem.
Theorem 2.1.13 (Equivalence of the definitions of continuity in R). A function f :
R → R is continuous in the usual sense if and only if it is continuous in the topological
sense.
Proof. We start with the first implication. Suppose that f is continuous. Let V be an
open set in R. Let x = f
−1
(
V
)
. Since V is open, there exists an ε such that B
ε
¡
f
(
x
)
¢
⊆
V by our previous characterization of open sets in R. We now use the continuity of
f . Since f is continuous, there exists δ >0 such that f
(
B
δ
(
x
))
⊆B
ε
¡
f (x)
¢
. Therefore
we have that f
(
B
δ
(
x
))
⊆V and thus B
δ
(x) ⊆ f
−1
(
V
)
. This shows f
−1
(
V
)
is open in
R because of our earlier characterization, again.
Suppose now that f is continuous in the topological sense. Let x ∈ R. We know
that for ε > 0, B
ε
(f (x)) is open in R (why?). Since f is continuous, f
−1
¡
B
ε
(f (x))
¢
is
open. Since f (x) ∈ B
ε
(f (x)), we have that x ∈ f
−1
¡
B
ε
(f (x))
¢
. Again, since f is con-
tinuous, there exists δ >0 such that B
δ
(x) ⊆ f
−1
¡
B
ε
(f (x))
¢
. This shows continuity in
the ε −δ sense (why?).
6
Once again, let us contrast continuity in R with our two examples from earlier,
the discrete and trivial topology.
EXAMPLE 2.1.14. Let X be an arbitrary set. Equip X with the discrete topology. Then
every function f : X → Y where Y is an arbitrary topological space is continuous.
Indeed, f
−1
(
V
)
⊆O
X
for all V ∈O
Y
, since every subset of X is in O
X
.
We leave the other example as an exercise.
EXERCISE 2.1.15. Characterize all the continuous functions f : X → Y , where X is
an arbitrary topological space and Y is an arbitrary set equipped with the trivial
topology O
Y
={Y ,∅}.
EXERCISE 2.1.16. Show that the composition of continuous functions between topo-
logical spaces is still continuous.
2.1.2 Connectedness and Path-Connectedness
An important question is that of invariants. A topological invariant is an abstract
property preserved via homeomorphisms. What this means is that if X ' Y , then
X and Y both possess the same invariants. A well-known example of an invariant
(that happens to be topological) is the Euler number of a polyhedron. The real use-
fulness of this statement lies in its contrapositive. Let ψ be a topological invariant
that computes a certain number. For instance, ψ(X ) = 2. If ψ(X ) 6= ψ(Y ), X can-
not be homeomorphic to Y . We will concern ourselves with three invariants in this
lecture: connectedness, path-connectedness and compactness. In this section, we
will define these properties and show they are topological invariants. We begin with
(path-)connectedness.
An interesting property of R is that intervals are ‘connected’. There is no hole be-
tween two elements of (a,b), unlike in Q where the irrationals puncture any ‘interval
in Q’. For instance, if we consider (0,2) ∈ R, every 0 < x < 2 is in (0,2), but
p
2 isn’t
in Q. There is a ‘hole’. In fact, there are many such holes if we restrict our interval
to Q, because Q is dense in R. Unlike continuity and convergence, connectedness
is expressed the same way in R and in general topological spaces. There are many
equivalent definition of connectedness, but we will restrict ourselves to only one for
now.
DEFINITION 2.1.17. A topological space X is said to be connected if X cannot be
made into a disjoint union of non-empty subsets A,B ∈ X .
In the special case where X =R with its usual topology, it is very clear that (0,1)∪
(2,5) isn’t connected. Intervals are however. We will not prove this fact extensively,
because we are more interested in general spaces than R. We do leave part of it as an
exercise however.
EXERCISE 2.1.18. Show that the open interval (a,b) is connected in R. Show that R
is connected.
Another topological invariant that is related to connectedness is path-connectedness.
7
DEFINITION 2.1.19. Let X be an arbitrary topological space. We say X is path-connected
if for every two points x, y ∈ X , there exists a continuous function ϕ :
[
0,1
]
→ X such
that ϕ(0) = x and ϕ(1) = y. Such a continuous function is called a path from x to y.
Proposition 2.1.20. A path-connected topological space is connected.
Proof. We prove this by contradiction. Suppose that X = A∪B, where A 6= B 6=∅ and
A∩B =∅. We choose a ∈ A and b ∈B. Since X is path-connected, there exists a path
ϕ between a and b. By continuity: ϕ
−1
(X ) = ϕ
−1
(A ∪B ) = ϕ
−1
(A) ∪ϕ
−1
(B) =
[
0,1
]
.
But
[
0,1
]
is connected, contradiction.
We now show those properties are invariants.
Theorem 2.1.21. Connected and path-connectedness are topological invariants.
Proof. We only prove path-connectedness, connectedness is left as an exercise. The
proofs are similar. Let X be a path-connected topological space, and let f : X →Y be
a homeomorphism. Let a,b ∈Y . Since f is a homeomorphism, there exists a
0
,b
0
∈ X
such that f (a
0
) = a, f (b
0
) =b. Since X is path-connected, there exists a path ϕ from
a
0
to b
0
. Since ϕ(0) = a
0
and ϕ(1) = b
0
, we have that f ◦ϕ(0) = a and f ◦ϕ(1) = b,
which shows that f ◦ϕ :
[
0,1
]
→Y is a path between a and b in Y .
EXERCISE 2.1.22. Prove that connectedness is a topological invariant.
2.1.3 Compactness
Compactness is a familiar property to anyone having taken a basic introduction to
analysis. The importance of compactness is hard to overstate. Very important re-
sults and constructions in real analysis depend crucially on the compactness of the
interval
[
a,b
]
. The Riemann integral is first defined over a compact interval before
we consider improper integrals over non-compact sets. Two very important results
state that a continuous function defined over an integral possesses a maximum, and
that it is uniformly continuous. The proofs of those statements can be found in any
books on basic real analysis. While there is no direct generalization of uniform con-
tinuity in topological spaces in general (there is one for a particular subset of them),
compactness in topological spaces is still important. We shall devote this section to
defining compactness in general spaces, prove that it is a topological invariant and
finally prove that over R, both definitions imply one another, which is the content of
the celebrated Heine-Borel theorem.
We first define what an open cover is.
DEFINITION 2.1.23. We say that a family of open sets {A
i
}
i∈I
is an open cover of X if
X ⊆
S
i=1
A
i
.
Equipped with this notion of covers, we can now state what it means for a topo-
logical space to be compact.
DEFINITION 2.1.24. X is a compact topological space if, for every open cover {A
i
}
i∈I
,
there exists a finite subcover. That is, for every {A
i
}
i∈I
, X ⊆
S
n
i=1
A
i
. Only finitely
many A
i
’s are required to cover X .
8
EXAMPLE 2.1.25. Any finite topological space is compact (why?).
It is not immediately obvious that the definition in R (closed and bounded set)
is equivalent to the more general definition. In fact, there are many definitions of
compactness. Another one is that a space is compact if every sequence has a con-
vergent subsequence. In the context of R, this is the Bolzano-Weierstrass theorem,
but in general spaces, this is called sequential compactness. It is not unusual that
great theorems turn into definitions when we go up a notch in abstractness.
Before showing the Hahn-Banach theorem, we demonstrate the invariance of
compactness.
Theorem 2.1.26 (Compactness is a topological invariant). Let X be a compact topo-
logical space, and let ϕ : X →Y be a homeomorphism. Then Y is compact.
Proof. Let F
Y
be an open cover for Y . Then f
−1
(
F
Y
)
:=
©
f
−1
(
U
)
|U ∈F
Y
ª
is an open
cover for X . By compactness of X , there exists a finite subcover of f
−1
(
F
Y
)
. But then
F
0
Y
={V
1
,...,V
n
} is a finite subcover of Y .
Now that we know compactness is a topological invariant, we can show some
interesting things. We leave the details to the reader.
EXERCISE 2.1.27. Show that R
n
is not homeomorphic to
[
0,1
]
n
. What about (0,1)
n
?
As we have done for continuity, it is important to check whether our definition of
compactness coincides with the usual concept of compactness on R. This important
theorem shows that the open cover definition of compactness is an accurate gener-
alization of the concept of being ‘closed and bounded’. The proof can be found in
any standard real analysis text and isn’t particularly illuminating, so we won’t men-
tion it here.
Theorem 2.1.28 (Heine-Borel). A set K ∈ R is compact if and only if it is closed and
bounded.
2.2 Categories
2.2.1 Categories and morphisms
As we said in the introduction, categories have become rather ubiquitous in mathe-
matics. We are going to define categories and give a couple of examples. You do not
need to understand every single example here to understand the material. Before
defining categories, we need to make a small remark on a finer point. As Russell’s
paradox shows, the notion of a set of all set can lead to some trouble. The famous
paradox discusses the set of all sets that are not members of themselves. To avoid
any kind of set-theoretical trouble, we will use classes. Classes are collections of sets.
For instance the class of all sets is well-defined. Do notice that this class is not a set!
DEFINITION 2.2.1. A category C is defined by:
1. A class Obj(C) of elements of C, called objects
9
2. A class hom
C
[
X ,Y
]
of maps between every X ,Y ∈Obj(C), called morphisms
3. A composition of morphisms ◦: hom
C
[
Y , Z
]
×hom
C
[
X ,Y
]
→hom
C
[
X , Z
]
such
that:
(a) The composition is associative: for f : A → B, g : B → C, h : C → D, we
have f ◦(g ◦h) =( f ◦g ) ◦h.
(b) Every element X ∈ Obj(C) has an identity morphism 1
X
: X → X such
that for every f : A →B we have 1
A
◦ f = f = f ◦1
B
.
This definition can seem quite general. We have some elements and some mor-
phisms between them. It turns out that this is general enough to encompass a large
collection of interesting mathematical structures, but specific enough that we can
state meaningful theorems about these structures. Let us see some example. We
leave it to the reader to show these examples are categories.
EXAMPLE 2.2.2. 1. The category Set consisting of the class of all sets along with
set functions between every object of the category.
2. The categories Grp, Ring, Field of all groups/rings/fields along with groups/rings/fields
homomorphisms.
3. The category Ab of all abelian groups with group homomorphisms.
4. The category Vect
k
of all vector spaces over the field k.
5. The category Top of topological spaces along with, you guessed it, continuous
functions.
6. The category Set* of pointed sets (sets with a base point), that is pairs of the
form (X , x) with x ∈ X and morphisms that take base points to base points.
7. The category Top* of pointed topological spaces, with continuous functions
that take base points to base points.
Notice how all these examples consist of a set along with some additional struc-
ture and/or a base point, and morphisms that preserve the structure between ob-
jects of the category. Categories which are built upon Set (like Grp or Top) with
additional structure are called concrete categories. Notice how Ab and Grp have the
same kind of morphisms. In a sense to be defined, Ab is a ‘subcategory’ of Grp, since
Ab consists of those group which also happen to be abelian. We will see later on how
we can define more precisely this idea of restricting structure on a category.
We saw that different mathematical structures have different isomorphisms be-
tween them. Isomorphism can be defined in general for categories as well.
DEFINITION 2.2.3. Let C be a category and let X , Y ∈Obj(C). We say that a morphism
f : X →Y is an isomorphism if there exists a morphism g : Y → X such that f ◦g =1
Y
and g ◦ f =1
X
. An endomorphism is a morphism f : X → X . An endomorphism that
happens to be an isomorphism is called an automorphism.
10
EXAMPLE 2.2.4. A homeomorphism is an isomorphism in the category Top. A bijec-
tion is an isomorphism in the category Set.
We have defined morphisms between objects of a category. It seems rather nat-
ural to ask whether there are functions between categories. Do notice that such
functions should be defined both for objects and morphisms of categories. Look-
ing back on our example with Ab and Grp, surely there should be a way to say that
we ignore the ‘commutativity’ of abelian groups to get back our group without the
abelian structure. And there is in fact a way to do this.
2.2.2 Functors and diagrams
DEFINITION 2.2.5. Let C
1
,C
2
be two categories. A functor F : C
1
→C
2
is a map that
assigns to each object X ∈C
1
an object F (X ) ∈C
2
and to each morphism f : X →Y
in C
1
a morphism F (f ) : F (X ) →F (Y ) such that:
1. F (1
X
) = 1
F (X )
(respects identities),
2. F ( f ◦ g ) = F ( f ) ◦F (g) for all morphisms f : X → Y and g : Y → Z in C
1
(respects composition).
We can now make clearer the idea of forgetting structure.
EXAMPLE 2.2.6. The group forgetful functor F : Grp → Set is the functor that sends
a group to its underlying set and homomorphisms to their underlying set function.
It ‘forgets’ the group structure. While not very formally defined, this is a common
example of a simple functor. A similar example can be given from Top to Set where
one forgets the topological structure.
In the last part of this lecture, we will see why category theory had its origins in
algebraic topology. Indeed, assigning an algebraic invariant to a topological space
is similar to defining a functor from Top to another (algebraic) category. Func-
tors in general can be used to describe a wide range of mathematical entities, from
something as simple as a topological invariant to certain quantum field theories
in physics. Before moving on to constructing new topologies, we need a few more
tools.
DEFINITION 2.2.7. We define the opposite category C
op
of a category C to be the cat-
egory formed with the same objects of C , but such that the morphisms are reversed,
i.e. we associate to any two objects X ,Y ∈C
op
the class hom
C
[
Y , X
]
, that is, maps of
the form f : Y → X and we fix Obj(C
op
) = Obj(C).
DEFINITION 2.2.8. A functor is said to be contravariant if it is of the form F : C
op
→
D for some categories C,D. It is similar to a normal functor (called a covariant func-
tor) except that it reverses the morphisms and the composition.
EXAMPLE 2.2.9. The functor ∗ : Vect
k
→ Vect
k
sends a vector space to its algebraic
dual space. Let V be a k-vector space. We send V to V
∗
:= {ϕ : V → k | ϕ is linear}
and we send each linear map f : V → W to its dual map f
∗
: W
∗
→ V
∗
,α 7→ α ◦ f .
Furthermore, we know that 1
∗
V
=1
V
∗
and that (f ◦g )
∗
= g
∗
◦ f
∗
which proves that ∗
is a contravariant functor.
11
The main goal of this lecture is to define common topologies using commuta-
tive diagrams. We are now equipped with the basic definitions of category theory
and topology required to make these constructions. We end this section by defining
diagrams and what it means for them to commute.
DEFINITION 2.2.10. A diagram is a collection of objects of a certain category and
maps (usually called arrows) between them. We say the diagram commutes if every
direct path along the arrows lead to the same result.
EXAMPLE 2.2.11. The following simple diagram commutes if g ◦ f =h:
A B
C
f
h
g
Commutative diagrams are really useful tools in category theory and homolog-
ical algebra. They are kind of similar to equalities in algebra, as in they define im-
portant properties of categories. For instance, many categories are defined as cate-
gories whose objects or morphisms satisfy certain commutative diagrams. There is
a more formal way to define diagrams, namely as a functor from an index category
to another category. We will not use this definition here because we want to focus
on topology and the intuitive definition works better for our purpose.
3 Topological constructions
We now come to the main part of the lecture. Equipped with category theory and
more specifically commutative diagrams, we define the basic topological construc-
tions that one can find in any elementary textbook on topology. Usually, these no-
tions will be defined differently. For instance, the subspace topology is defined with
intersections of open sets. I strongly believe that the ‘categorical’ definition (with
a commutative diagram) is clearer in that it shows what’s happening behind the
scenes. It is not clear what the product and subspace topology have in common,
but once they are defined in terms of diagrams, the connection is immediately ob-
vious. We begin by defining two very important general constructions that are dual
to one another.
3.1 Final and Initial topology
The idea behind the initial topology is to ‘pull back’ the topology on a family of
spaces to an arbitrary set, thereby constructing a topological space (X ,O ). To this
effect, we will use a family of functions.
DEFINITION 3.1.1 (Initial topology). Let X be an arbitrary set, let {Y
i
}
i∈I
be an in-
dexed family of topological spaces and let {f
i
}
i∈I
be a family of function f
i
: X →Y
i
.
The initial topology O
I
on X is the coarsest topology on X such that each function
f
i
is continuous.
12
While this definition is rather general and abstract, it is exactly what we need
to precisely define some important examples of topologies. Before doing that, we
introduce its dual notion, the final topology. Notice the ‘arrow reversal’ that is char-
acteristic of a dual construction.
DEFINITION 3.1.2 (Final topology). Let X be an arbitrary set, let {Y
i
}
i∈I
be an indexed
family of topological spaces and let { f
i
}
i∈I
be a family of function f
i
: Y
i
→ X . The
final topology on X is the finest topology such that each function f
i
is continuous.
The difference in these two definitions is the domains/codomains of the func-
tions. This time, the idea behind the final topology is to ‘push forward’ the topology
on X via the topology of each Y
i
and of our family of functions. We now use com-
mutative diagrams to describe how topological spaces equipped with initial/final
topology behave with respect to continuous maps from/to other morphisms.
Theorem 3.1.3 (Characteristic property of the initial topology). Let X be a topolog-
ical space with the initial topology and let Z be a topological space. The function
ϕ : Z → X is continuous if and only if f
i
◦ϕ is continuous for all i ∈ I .
X Y
i
Z
f
i
ϕ
f
i
◦ϕ
Likewise, we have a characteristic property for the final topology.
Theorem 3.1.4 (Characteristic property of the final topology). Let X be a topological
space with the initial topology and let Z be a topological space. The function ϕ : X →
Z is continuous if and only if ϕ ◦ f
i
is continuous for all i ∈ I .
Y
i
X
Z
f
i
ϕ◦f
i
ϕ
These properties will naturally transfer over to the topologies we will construct
in the next section.
3.2 Products
3.2.1 First attempt: The Box Topology
We finally begin constructing various examples of important topologies. The first is
that of a Cartesian product of topological spaces. The most intuitive case is probably
that of R
2
=R ×R. We have already defined a topology on R. We wish to extend this
topology to R
2
. A natural generalization is to take Cartesian products of open sets in
R. Namely, we set O
R
2
=O
R
×O
R
:={U ×V |U ,V ∈O
R
}.
13
EXERCISE 3.2.1. Prove that R
2
equipped with the topology above is a topological
space.
While simple, this generalization works well for R
2
. In fact, it works well for any n
by using a similar argument that you gave in the exercise. But there is nothing special
about the topology on R with respect to this product, indeed we have never used any
properties specific to R to define this topology. This suggests a more generalized
definition of this topology.
DEFINITION 3.2.2 (The Box Topology). Let (X
i
,O
i
) be a family of topological spaces.
Let X =
Q
i=1
X
i
denote the Cartesian product of our topological spaces. We can
define a topology on this product called the box topology by setting O
X
:={
Q
i=1
U
i
|
U
i
∈O
i
}.
It is not difficult to prove that the box topology is, in fact, a topology. As you may
have guessed from the title of this subsection, this isn’t the only possible topology
we can put on our product. When we take finite products, this topology is totally
reasonable and our intuition is still correct. The problems start occurring when we
take infinite products.
DEFINITION 3.2.3. We define the set of real sequences R
ω
to be R
ω
:=
Q
∞
i=1
R
i
.
If we equip R
ω
with the box topology, we soon get some very concerning prob-
lems. The box topology contains many open sets, too many for basic results to hold
true. The following simple example shows that continuity in the components does
not mean continuity of the function as a whole.
EXAMPLE 3.2.4. Let R
ω
be the space of real sequences with the box topology. Con-
sider the function f : R →R
ω
that is the identity in every component: f : x 7→(x, x, x,...).
It is clear that the component functions f
i
: x 7→x are continuous because the iden-
tity is continuous on R. Yet this function is not continuous. Suppose f is continuous.
We consider the open set
U =
∞
Y
i=1
µ
−
1
i
,
1
i
¶
.
f (0) = (0, 0,0,... ) ∈ U . By continuity of f , there should exist a small neighborhood
(−ε,ε) with ε >0 such that (−ε, ε) ⊂ f
−1
(
U
)
. But this implies that
f
³
ε
2
´
=
³
ε
2
,
ε
2
,
ε
2
,...
´
⊂U
which is clearly false since
ε
2
>
1
n
for n >
2
ε
.
This little example shows just how bad our topology really is for infinite products.
The problem is that there are way too many open sets. We need a coarser topology.
This example should underlie why initial and final topology are important. They are
the coarsest/finest topology such that continuity can be described via component
functions, unlike the box topology we have constructed in this section. This begs for
an alternate definition of a topology on a product.
14
3.2.2 Second attempt: The Product Topology
This time around, we will construct a product topology on
Q
∞
i=1
X
i
via an initial
topology.
DEFINITION 3.2.5 (Canonical projections). Let X =
Q
∞
i=1
X
i
. We define the canonical
projections to be functions π
i
: X → X
i
that send an element x =(x
1
, x
2
,..., x
i
,...) to
the i-th coordinate x
i
.
The canonical projections define a family of functions indexed by I , associated
to a family of topological spaces X
i
. We now use these projections to define our
topology.
DEFINITION 3.2.6. The product topology is the initial topology with respect to the
canonical projections. That is, it is the coarsest topology on X such that each π
i
is
continuous. We can describe it with the following commutative diagram:
X X
i
Y
π
i
ϕ
π
i
◦ϕ
ϕ is continuous if and only if its composition with each component is continuous.
This topology is much better behaved than the box topology. In the finite case,
they coincide. The reader is free to try to prove that the function defined in 3.2.4 is
now continuous in this topology. The open sets in a space equipped with the prod-
uct topology are of the form U = {
Q
i=1
U
i
|U
i
∈ X
i
and U
i
6= X
i
for only finitely many i}.
3.3 Subspaces
The case of subspaces is easier to handle. There are actually two equivalent ways to
define the subspace topology. The first one is intuitive and the usual definition in
most standard introductions to topology.
DEFINITION 3.3.1 (Subspace Topology: I). Let (X ,O
X
) be a topological space, and let
A be a subset of X . The subspace topology on A is defined as O
A
:={U ∩A |U ∈O
X
}.
EXERCISE 3.3.2. Show this is a topology.
What is sometimes not known to students new to topology is that this definition
is precisely the initial topology with respect to the inclusion map.
DEFINITION 3.3.3. The inclusion map is the map ι : A → X where A ⊆ X that sends
A to itself in X .
We now give our definition of the subspace topology.
DEFINITION 3.3.4 (Subspace Topology: II). The subspace topology is the coarsest
topology on A such that the inclusion map is continuous. We can describe it with
the following diagram:
15
X
Z Y
ϕ◦ι
ϕ
ι
Thus a function to the subspace A ⊆ X is continuous if and only if it is continuous
in X when composed with the inclusion map.
3.4 Dual of the product topology: disjoint unions
We want to equip disjoint unions of topological spaces with a topology. This time,
we will use a final topology to do this.
DEFINITION 3.4.1. Let X =
F
i=1
X
i
be the disjoint union of the indexed family {X
i
}
i∈I
.
We define the canonical injections φ
i
: X
i
→ X defined by φ(x) =(x,i).
We will use these injections to define our topology.
DEFINITION 3.4.2. Let X =
F
i=1
X
i
be the disjoint union of the indexed family {X
i
}
i∈I
.
The disjoint union topology is the finest topology on X such that the canonical injec-
tions stay continuous. This topology can be characterized by the following universal
property: If Y is a topological space, and f
i
: X
i
→ Y is a continuous function for
each i ∈ I , then there exists a unique continuous map f : X → Y such that the fol-
lowing diagram commutes:
X
X
i
Y
f
φ
i
f
i
3.5 Dual of the subspace topology: quotient spaces
The intuition behind quotient spaces is altogether more involved than for subspaces,
or product. The geometric idea is that we want to identify (or glue) certain points to-
gether, to form a smaller space. To do that, we will define an equivalence relation on
our space, and then take our space to be the equivalence classes (which necessarily
partition the entire space). This is in fact similar to what we do with groups when we
define quotient groups. The equivalence relation was simply hidden in the cosets.
Remember from set theory that if x ∈[y] and x ∈[z], then [y] =[z].
DEFINITION 3.5.1. Let X be a topological space and let ∼be an equivalence relation
on X . We define the quotient space X / ∼ to be the set X / ∼:={
[
x
]
|x ∈ X }.
Our goal in this section is to define a topology on X / ∼. To this extent, we will use
a special map and define the topology on X / ∼ to be the final topology with respect
to this map.
DEFINITION 3.5.2. Let X / ∼ be a quotient space. We define the canonical quotient
map to be f : X → X / ∼, x 7→[x].
16
DEFINITION 3.5.3. Let X / ∼ be a quotient space. The quotient topology on X / ∼ is
the final topology with respect to the canonical quotient map. The quotient space
and the canonical quotient map are characterized by the following universal prop-
erty: Let g be a continuous function from X →Y such that x ∼ y implies g (x) =g (y)
for all x, y ∈ X . Then, there exists a unique continuous map f : X / ∼→ Y such that
g = f ◦q. We say that g descends to the quotient.
EXAMPLE 3.5.4. Let D
2
denote the two-dimensional disk D
2
:= {x ∈ R
2
| k xk ≤ 1}.
Denote its boundary by ∂D
2
. Then D
2
/∂D
2
'S
2
.
EXAMPLE 3.5.5. Let S
1
:={x ∈ C |kxk = 1} denote the unit circle. We have that R/Z '
S
1
by the homeomorphism x 7→ e
2πi x
.
3.6 Weak-
∗
topology
DEFINITION 3.6.1. Let K be a field equipped with a topology. We say that V is a
normed vector space if it is a vector space over K equipped with a norm. The topology
induced by the norm will be called the strong topology on V .
From now on, K stands for either R or C.
EXAMPLE 3.6.2. We can equip R
n
with the usual Euclidean norm. It is clear that R
n
is a vector space over R.
Our goal in this section will be to define another topology on V that will be useful
in functional analysis.
DEFINITION 3.6.3. Let V be a vector space over K . The continuous dual space of V is
the vector space of bounded linear functionals V
∗
:=
©
ϕ : V →K |ϕ is linear and bounded
ª
.
The double dual space is then defined as V
∗∗
:=
©
ϕ : V
∗
→K |ϕ is linear and bounded
ª
.
We now recall from functional analysis that there is an injection into the double
dual space (an isomorphism if V is finite-dimensional). We start by constructing the
evaluation map that takes a linear functional and applies it to elements of V .
DEFINITION 3.6.4. Let V be a normed vector space. The evaluation map ev
v
: V
∗
→
K is the linear map ϕ 7→ ϕ(v). We then define the double-dual injection to be the
map Ψ
v
: V →V
∗∗
such that v 7→ev
v
, for all v ∈V .
It is with respect to these evaluation maps that we are going to define the weak-
∗
topology on V
∗
.
DEFINITION 3.6.5 (Weak-
∗
topology). The weak-
∗
topology on V
∗
is the coarsest
topology on V
∗
such that every map ev
v
stays continuous. In other words, it is the
initial topology with respect to the maps ev
v
. This topology is useful for multiple
reasons, one of which is the celebrated Banach-Alaoglu theorem from functional
analysis which states that the closed unit ball of V
∗
is compact in the weak-* topol-
ogy.
17
