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-
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