7  The notion of a topology

In our discussion of Euclidean space and finite-dimensional Hilbert spaces in Section 4.4, we had an inner product at our disposal. The inner product could be used to define a norm, that measured length. Appealing to the linear structure of the space, we could measure distances, \(d(x,y) = \|x - y\|\).

A distance measure gives us the ability to talk about closeness between points, i.e., a topology. In the field of topology, one makes these concepts abstract. In particular, we get axioms for open sets. These sets are used, for example, to define various concepts of functions \(f : U\to V\) being continuous.

Definition 7.1 (Topological space) Let \(X\) be a set. A topology \(\tau\) on \(X\) is a collection of subsets of \(X\), called open sets, satisfying the following axioms:

• The empty set \(\emptyset \in \tau\) Empty set is open

• \(X \in tau\) The space itself is open

• If \(\{U_i \mid i \in \text{some index set}\}\) is an arbitrary collection of elements of \(\tau\), then \[\bigcup U_i \in \tau.\]Arbitary unions are open

• If \(\{U_i \mid i \in \{1,2,\cdots,n\}\}\) is a finite collection of elements of tau, then \[\bigcap U_i \in \tau\]Finite intersections are open

The pair \((X,\tau)\) is called a topological space.

A subset \(C\subset X\) is said to be closed in \(X\) if its complement \(C^\complement = X \setminus C\) is open.