Neighborhoods

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Section Neighborhoods

We are familiar with the idea of open intervals in \(\R\text{.}\) We next introduce the idea of an open neighborhood of a point and characterize continuity in terms of neighborhoods. This is the next step in developing the notation of continuity in topological spaces.

The open ball \(B(a, \delta)\) in a metric space \((X,d)\) is also called the \(\delta\)-neighborhood around \(a\text{.}\) A neighborhood of a point can be thought of as any set that envelops that point.

Definition 7.2.

Let \((X, d_X)\) be a metric space, and let \(a \in X\text{.}\) A subset \(N\) of \(X\) is a neighborhood of \(a\) if there exists a \(\delta \gt 0\) such that \(B(a, \delta) \subseteq N\text{.}\)

Example 7.3.

In \(\R\) with the Euclidean metric, the set \(\R^+\) (the positive real numbers) is a neighborhood of \(a=1\) because the open ball \(B(1,0.5)\) is completely contained in \(\R^+\text{.}\)
In \(\R\) with the Euclidean metric, the set \(\Z\) is not a neighborhood of \(a=1\) because any open ball centered at \(a=1\) will contain some non-integers.
In \(\R\) with the discrete metric, the set \(\Z\) is a neighborhood of \(a=1\) because the open ball \(B(a,1) = \{a\}\text{.}\)

As another example, the open ball \(B(a, \delta)\) is a neighborhood of \(a\text{.}\) We can say even more about open balls.

Activity 7.2.

Let \((X, d)\) be a metric space, let \(a \in X\text{,}\) and let \(\delta \gt 0\text{.}\) In this activity we ask the question, is \(B(a, \delta)\) a neighborhood of each of its points?

(a)

Let \(b \in B(a, \delta)\text{.}\) What do we have to do to show that \(B(a, \delta)\) is a neighborhood of \(b\text{?}\)

(b)

Use Figure 7.4 to help show that \(B(a, \delta)\) is a neighborhood of \(b\text{.}\)

Figure 7.4. \(B(a, \delta)\) as a neighborhood of \(b\text{.}\)

(c)

Is the converse true? That is, if a set is a neighborhood of each of its points, is the set an open ball? No proof is necessary, but a convincing argument is in order.

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