Some Point-set Topology: Part 1

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

 

Topological spaces

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 \mathbb{R} or the d-dimensional euclidean space \mathbb{R}^d. 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 x_0 a function f:\mathbb{R}\to \mathbb{R}  that satisfy the following property:
for all \varepsilon>0, there exists some \delta>0 such that, if x is ]x_0-\delta,x_0+\delta[, its image f(x) is in ]f(x_0)-\varepsilon,f(x_0)+\varepsilon[. We say that f:\mathbb{R}\to\mathbb{R} is continuous when it is continuous at every point x\in\mathbb{R}. 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 U\subset \mathbb{R} is open if it satisfy the following property: for each x in U, there exist a real number \varepsilon>0 such that the interval ]x-\varepsilon, x+\varepsilon[\subset U. It is then easy to see that, with this definition,

  1.  the whole real line \mathbb{R} and the empty set \emptyset are open;
  2. the union of a collection of open sets is open;
  3. the intersection of a finite number of open sets is open.

By playing around a bit with this definition, we can also see that f:\mathbb{R}\to \mathbb{R} is continuous if, and only if, the preimage f^{-1}(U) of any open set U \in \mathbb{R} is open.

In the above paragraph, the intervals (and thus the order relation on \mathbb{R}) 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 \mathbb{R}, it is enough to specify the open subsets  More precisely, we call topological space a set X, together with a collection \tau of subsets of X such that:

  1. X and \emptyset are in X;
  2. if (U_i)_{i\in I} is a family of subsets in \tau, \bigcup_{i\in I}U_i\in \tau;
  3. if (U_i)_{i\in I} is a finite family of subsets in \tau, \bigcap_{i\in I}U_i\in \tau;

In the second axiom, the set of indexes I need not be finite, or even countable. We say that subsets of X that are in \tau are open, and that subsets of X whose complement are in \tau closed. Now, if (X_1,\tau_1) and (X_2,\tau_2) are two topological spaces, we say that a function f:X_1\to X_2 is continuous if, for all U\in \tau_2, the preimage f^{-1}(U) is in \tau_1.

If (X,\tau) is a topological space and Y\subset X, we define a topology on Y by specifying the open sets in the following way: a set V\subset Y is open when there exists U \in \tau such that V=U\cap Y (Exercise: Show that this indeed defines a topology). Of course U need not be unique. The set Y with this topology is called a topological subspace of (X,\tau).

We often need to define a topology without specifying all the open sets. With that in mind, we call \beta \subset \tau a basis for (X,\tau) when any open set is the union of sets in $\beta$. For instance, by the very definition of open sets in \mathbb{R} used at the beginning, the collection containing all the sets ]x-\varepsilon,x+\varepsilon[ with x\in \mathbb{R} and \varepsilon>0 is a basis.

Finally, if x\in X, we often want to restrict our attention to points that are “close” to x. We say that a set N (not necessarily open) is a neighborhood of x when there exist U\in \tau such that x\in U \subset N. We say that a collection \beta_x of subsets of X is a neighborhood basis at x if each member of \beta_x is a neighborhood of x, and if, for any neighborhood N of x, there exists N'\in \beta_x such that N'\subset N. For instance,  if x \in \mathbb{R}, the intervals ]x-\varepsilon, x+\varepsilon[, with \varepsilon>0, are a neighborhood basis at x 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 (X_1,\tau_1) and (X_2,\tau_2) be topological spaces. We say that a function
f:X_1\to X_2 is continuous at x if, for every neighborhood N_2 of $f(x)$, the exist a neighborhood N_1 of x such that f(N_1)\subset N_2.  As an exercise, the reader should check the following result, which show that everything is consistent.

Proposition
The function f:X_1\to X_2 is continuous at every point of X_1 if, and only if, it is continuous (in the sense that the preimage of any open set is an open set).

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