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In my real analysis book, neighbourhood of a point is defined as

Let $c\in \mathbb{R}$. A subset $S\subset \mathbb{R}$ is said to be a neighbourhood of $c$ if there exists an open interval $(a,b)$ such that $c\in (a,b)\subset S$.

And then the open set is defined in terms of neighbourhood. But as open interval is an open set, so it looks like here we are defining neighbourhood through open sets and then vice versa. How to get out of this loop?

P.S. I don't have any background knowledge of general topology. I heard that there can be two ways to approach the definition of topological space,

  1. By defining neighbourhood in terms of open sets or,
  2. By defining open sets in terms of neighbourhood.

However, these definitions are very rigorous and counterintuitive for me. Also if learning general topology is the only way to understand independent definitions of neighbourhood and open sets, then can you suggest any good book or other material from which a second-year undergraduate may start.

Soham Sarkar
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    Don't be confused by wording. The fact that you call the intervals of the form $(a,b)$ "open intervals" doesn't mean that they are necessarily "open sets" in the sense of the above definition. (My favourite saying when I want to point to people to not pay too much attention at wording is that "a hot dog is not a dog".) This time, however, terminology is reasonable, because you can then easily (actually instantly!) prove that an open interval is an open set in the sense of the above definition. No circularity here. –  Apr 12 '21 at 20:49

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A topology can be defined in several equivalent ways: using a family of subsets sets that obeys three axioms is the most common way to start, these sets are then called the open sets.

Another way that is quite old historically is to specify for each point in $X$ a set of subsets $\mathcal{N}(x)$ that all contain $x$ and satisfy a quite different set of axioms.

We can define concepts like continuity, connectedness, compactness and everything of interest either in terms of these neighbourhoods of points or in terms of open sets.

Once we have open sets we can define neighbourhood systems and vice versa once we have neighbourhood systems we can define open sets: a set $N$ is a neighbourhood of $x$ iff there is an open set $O$ such that $x \in O \subseteq N$ (and if the open sets obey the open set axioms this definition obeys the neighbourhood axioms) and a set $O$ is open iff for all $x \in O: O \in \mathcal{N}(x)$ (open = a neighbourhood of each of its points) and the same applies: a neighbourhood assignment gives a valid topology in terms of open sets. Also: these maps are mutual inverses (hence they are equivalent ways to view the same space).

So there really is no circularity: they’re just different ways to look at topological concepts; more globally (open sets) or locally (neighbourhoods). They give the same concepts and results eventually.

Henno Brandsma
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