Posted on 26th February 2019

We saw before that on a set \( X \) we can specify a (unique) topology by, for each \( x\in X \), specifying a collection of sets \( U_x \) which will satisfy that:

- Each \( V\in U_x \) contains \( x \) and will be open;
- Every open \( C \ni x \) will be such that there is \( V\in U_x \) with \( V\subseteq C \),

if and only if we have the conditions that:

- Given \( A_1,\cdots,A_n\in U_x \), there is \( A\in U_x \) with \( A\subseteq A_1\cap\cdots\cap A_n \);
- Given \( B\in U_y \) with \( x\in B \), there is \( A\in U_x \) with \( A\subseteq B \).

However, we might ask: what is the advantage of specifying the "basic open sets" about each point, rather than just specifying a base for the topology?

One advantage comes when you consider subspaces. Let \( Y\subseteq X \) be a subset. There are then two natural ways to construct a topology on Y:

- Use the subspace topology; that is, the open subsets of \( Y \) are exactly the sets \( U\cap Y \) for \( U \) open in \( X \);
- For each \( y\in Y \) let \( V_y = \{ A\cap Y : A\in U_y\} \) and generate a topology using \( V_y \) as the basic open neighbourhoods of \( y \).

Do these give the same topology? We will see that the answer is yes. Firstly, it is clear that every member of \( V_y \) is open in the subspace topology. Conversely, given \( V = U\cap Y \) open in the subspace topology, for \( y\in V \), there is \( A\in U_y \) with \( A\subseteq U \), so \( B=A\cap Y \) is such that \( B\in V_y \) with \( B\subseteq V \). Thus \( V \) is in the topology generated by the \( V_y \).

In many situations, the sets \( V_y \) will occur very naturally. For example, the topology induced by a metric has as basic open sets about \( x \) the open balls centred at \( x \). The above reasoning now immediately shows that the subspace topology on \( Y\subseteq X \) will agree with the topology given by restricting to metric to \( Y \) and then considering \( Y \) as its own metric space.

This post was originally motivated by Fell's topology on a set of (equivalence classes) of unitary representations of a locally compact group \( G \). A basic open set about a representation \( \pi \) is \( W=W(\pi,\varphi_i,Q,\epsilon) \) where \( \epsilon>0 \), \( Q\subseteq G \) is compact, and \( (\varphi_i)_{i=1}^n \) are functions of positive type associated to \( \pi \), and \( \rho\in W \) exactly when we can approximate each \( \varphi_i \), up to \( \epsilon \) error on \( Q \), by sums of functions of positive type associated to \( \rho \).

Clearly the first (intersection related) condition holds for this family. The second condition seems trickier: if \( \rho\in W \), then we wish to find \( W' = W(\rho,\varphi_j',Q',\epsilon') \) such that if \( \rho'\in W' \) then \( \rho'\in W \). Really this is a triangle inequality argument. \[ |\varphi_i(g) - \sum_k \phi_{i,k}(g)|<\epsilon \qquad (g\in Q) \] where each \( \phi_{i,k} \) is associated to \( \rho \). (This follows as \( \rho\in W \)). As \( Q \) is compact, we can decrease \( \epsilon \) slightly and still have a true inequality: this allows us some wiggle room. Now if each \( \phi_{i,k} \) is very well approximated on \( Q \) by sums of functions of positive type associated to \( \rho' \), then the \( \varphi_i \) can also be \( \epsilon \) approximated by such functions. Thus \( \rho'\in W'(\rho,\phi_{i,k},Q,\epsilon') \) does imply \( \rho'\in W \), for some small \( \epsilon'>0 \).

So this topology interacts nicely with subspaces. In particular, the definition works well for subsets of the (proper class) collection of all representations.