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**Ackbeet** That does seem a bit circular. Any chance you could scan in and post the pages that have those two definitions?

Off-hand, I would say that, typically, open sets are defined as the members of a topological space as follows: a **topological space** is a set, X, together with a collection, T, of subsets of X, called "open" sets, which satisfy the following rules:

T1. The set X itself is "open"

T2. The empty set is "open"

T3. Arbitrary unions of "open" sets are "open"

T4. Finite intersections of "open" sets are "open".

- Martin D. Crossley, *Essential Topology*, p. 15.

You can define open sets in $\displaystyle \mathbb{R}$ and higher-dimensional spaces (really, any metric space) by looking at open balls, which is a more fundamental concept that open sets. In that setting, I would define a neighborhood, N, of x as a set such that there exists an open ball containing x that is itself contained in N. The concept of open balls does not depend on open sets, but it does need a metric.

The Crossley book, incidentally, is one I would recommend. Explanations seem very straight-forward to me.