Statement of intent
I have picked up differential geometry and topology where I left it three years ago. It keeps popping up in my research, and I have decided that I should study it seriously, with a precise treatment of the foundational material. I will record my progress here, in order to motivate me and to force me to be as complete and precise as I can. For the moment, I am using Foundations of Differentiable Manifolds and Lie Groups, by Frank W. Warner, and Topology and Geometry, by Glen E. Bredon. I also use two french books, Cours de Topologie, by Gustave Choquet, and Introduction aux Variétés Différentielles, by Jacques Lafontaine.
I will start with a review of some point-set topology. I will not attempt to be complete on this topic, I am mainly interested in learning the concepts that I might need to study manifolds. I will however try to define the objects that I use and motivate these definitions
When we first study functions of one or several real variables, we quickly define continuity and prove results dealing with continuous function, such as that the intermediate value theorem or the extreme value theorem. Continuous functions are the basic objects of analysis. If we go further in our study, we encounter function that map the elements of a set to the elements of another set, where the set need not be subsets of the real line or the -dimensional euclidean space . If we want to keep talking about continuous functions, we need to specify the simplest mathematical structure that allows us to do so.
Let us stay on the real line for the moment. We have learned in our calculus lectures to call continuous at a function that satisfy the following property:
for all , there exists some such that, if is , its image is in . We say that is continuous when it is continuous at every point . This definition is not very suitable to generalization, mostly because the definition of intervals relies on the fact that real number can be ordered. But it can be reformulated in a much more powerful way, at the price of increased abstraction.
Let us say that a subset is open if it satisfy the following property: for each in , there exist a real number such that the interval . It is then easy to see that, with this definition,
- the whole real line and the empty set are open;
- the union of a collection of open sets is open;
- the intersection of a finite number of open sets is open.
By playing around a bit with this definition, we can also see that is continuous if, and only if, the preimage of any open set is open.
In the above paragraph, the intervals (and thus the order relation on ) were used to define the open set, but after that we were able to define the continuity of a function purely in terms of open sets. This suggests that, to talk about continuity on sets that are more general than , it is enough to specify the open subsets More precisely, we call topological space a set , together with a collection of subsets of such that:
- and are in ;
- if is a family of subsets in , ;
- if is a finite family of subsets in , ;
In the second axiom, the set of indexes need not be finite, or even countable. We say that subsets of that are in are open, and that subsets of whose complement are in closed. Now, if and are two topological spaces, we say that a function is continuous if, for all , the preimage is in .
If is a topological space and , we define a topology on by specifying the open sets in the following way: a set is open when there exists such that (Exercise: Show that this indeed defines a topology). Of course need not be unique. The set with this topology is called a topological subspace of .
We often need to define a topology without specifying all the open sets. With that in mind, we call a basis for when any open set is the union of sets in $\beta$. For instance, by the very definition of open sets in used at the beginning, the collection containing all the sets with and is a basis.
Finally, if , we often want to restrict our attention to points that are “close” to . We say that a set (not necessarily open) is a neighborhood of when there exist such that . We say that a collection of subsets of is a neighborhood basis at if each member of is a neighborhood of , and if, for any neighborhood of , there exists such that . For instance, if , the intervals , with , are a neighborhood basis at for the topology defined at the beginning.
This allows us to give a general definition of the concept of continuity at a point, from which we started. Let and be topological spaces. We say that a function
is continuous at if, for every neighborhood of $f(x)$, the exist a neighborhood of such that . As an exercise, the reader should check the following result, which show that everything is consistent.
The function is continuous at every point of if, and only if, it is continuous (in the sense that the preimage of any open set is an open set).