Let be the product of with itself, let be an open tubular neighbourhood of the diagonal in .

It is defined to be the largest (closed) subset of X for which every open neighbourhood of every point of the set has positive measure.

Now by computing derivatives at 0, the union contains an open neighbourhood of 1.

A locally normal space is a topological space where every point has an open neighbourhood that is normal.

However, if the function f is continuously differentiable in an open neighbourhood of a fixed point x, and , attraction is guaranteed.

The basis of open neighbourhoods of 0 in R is given by the powers I, which are nested and form a descending filtration on R:

If is open it is called an open neighbourhood.

Here, a network is a family of sets, for which, for all points and open neighbourhoods , there is a for which .

It is symmetric since which by continuity of the inverse is another open neighbourhood of the identity.

It is transitive since where and are open neighbourhoods of the identity such that ; such pairs exist by the continuity of the group operation.