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Section 2.1 Topology

Given two sets \(A\) and \(B\text{,}\) an important object is the set of maps \(f:A\to B\text{.}\) The set of all maps from \(A\) and \(B\) is a purely combinatorial object. This set is often denoted by "\(B^A\)" since, when \(A\) and \(B\) are finite sets with respectively \(a\) and \(b\) elements, the number of maps from \(A\) to \(B\) is precisely equal to \(b^a\) (see Exercise 2.7.1). Analogously, the set of all subsets of a given set \(A\) is denoted by \(2^A\) since each subset \(S\subset A\) can be though as a map \(f_S:A\to\{0,1\}\) whose value is 1 on the elements of \(S\) and 0 otherwise.

The set \(B^A\) is way too large for practical purposes, especially when \(A\) and \(B\) are uncountable. It took centuries to spot a quite important type of subset of \(B^A\) whose elements can be studied in a more effective way. These are the class of continuous functions. In order to introduce continuity, that intuitively means functions whose output "changes not much" when the input "changes not much", one must give a meaning to "being close to". This is achieved by fixing a topology on A and B.
Definition 2.1.1. Topology.
A topology on a set A is a collection \(\cU\) of subsets of \(A\) such that:
  1. \(\emptyset,A\in\cU\text{;}\)
  2. \(\cU\) is closed under union;
  3. \(\cU\) is closed under finite intersection.
The sets of \(\cU\) are the open sets of \(A\text{.}\) Open sets are the tool used to define continuity and convergence.

A quite important property for a topology is to be able to separate points, namely whether, for each pair of distinct points \(x,y\in A\text{,}\) there are disjoint open sets \(U,V\) such that \(x\in U\) and \(y\in V\text{.}\)
Definition 2.1.2. Hausdorff space.
A topological space \(A\) is Hausdorff if there are enough open sets to separate each pair of distinct points.
Now that we have open sets, we can define continuous functions and converging sequences.
Definition 2.1.3. Continuous function.
Let \(A\) and \(B\) be two topological spaces with topologies respectively \(\cU_A\) and \(\cU_B\text{.}\) A function \(f:A\to B\) is continuous if \(f^{-1}(U)\in\cU_A\) for every \(U\in\cU_B\text{.}\)
Continuity is a strong constraint for a function. For instance, if \(f:\bR^n\to\bR\) is continuous and \(f(x_1)\neq f(x_2)\text{,}\) we know that necessarily for every \(c\) between \(f(x_1)\) and \(f(x_2)\) there must be at least an \(a\in\bR^n\) such that \(f(a)=c\text{.}\)
Definition 2.1.4. Converging sequence.
A neighborhood of a point \(a\in A\) is a set \(N\) that contains an open set \(U\in\cU_A\) which contains \(x\text{.}\)

Let \(a_n\) be a sequence in \(A\text{.}\) We say that \(a_n\) converges to \(a_0\) if, for every neighborhood \(N\) of \(a\text{,}\) almost all of the \(a_n\) are in \(N\text{.}\)
The collection \(\cU=2^A\) of all subsets of \(A\) is clearly a topology foir \(A\) and it is called the discrete topology. Each point in open (and closed) in the discrete topology. Hence, this topology is Hausdorff and with it "every point is infinitely distant from every other point", since each point has a neighborhood (itself!) where there is nothing else but the point itself. If \(A\) has the discrete topology, every map \(f:A\to B\) is continuous and only eventually constant sequences converge.
At the other end is the indiscrete topology, where \(\cU=\{\emptyset,A\}\text{.}\) In this case, all points are "arbitrarily close" to each other. In particular, this topology is far from being Hausdorff: no pair of distinct points is separated! Unless \(A\) has less than two points, no sequence converge here and only constant functions \(f:A\to\bR\) are continuous.
The following is an elementary but important observation: