Neighbourhoods

This situation often occurs. We need to be able to talk about a function near a point: in the above example, we don't want to worry about the singularity at x = - a when we are discussing the one at x = a (which is actually much better behaved). If we only look at the points distant less than d for a, we are really looking at an interval (a - d, a + d ); we call such an interval a neighbourhood of a. For traditional reasons, we usually replace the distance d by its Greek equivalent, and speak of a distance $\delta$. If $\delta$ > 0 we call the interval (a - $\delta$, a + $\delta$) a neighbourhood (sometimes a $\delta$ - neighbourhood) of a. The significance of a neighbourhood is that it is an interval in which we can look at the behaviour of a function without being distracted by other irrelevant behaviours. It usually doesn't matter whether $\delta$ is very big or not. To see this, consider:

Exercise 1.10   Show that an open interval contains a neighbourhood of each of its points.

We can rephrase the result of Ex 1.7 in this language; given l$\ne$m there is some (sufficiently small) $\delta$ such that we can find disjoint $\delta$ - neighbourhoods of l and m. We use this result in Prop 2.6.