Skip to main content

Topology: An Inquiry-Based Approach

Steven Schlicker

πŸ”—

Section Neighborhoods

πŸ”—
We are familiar with the idea of open intervals in R. 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.
πŸ”—
The open ball B(a,Ξ΄) in a metric space (X,d) is also called the Ξ΄-neighborhood around a. A neighborhood of a point can be thought of as any set that envelops that point.
πŸ”—

Definition 7.2.

πŸ”—
Let (X,dX) be a metric space, and let a∈X. A subset N of X is a neighborhood of a if there exists a Ξ΄>0 such that B(a,Ξ΄)βŠ†N.
πŸ”—

Example 7.3.

πŸ”—
  • πŸ”—
    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+.
  • πŸ”—
    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.
  • πŸ”—
    In R with the discrete metric, the set Z is a neighborhood of a=1 because the open ball B(a,1)={a}.
πŸ”—
As another example, the open ball B(a,Ξ΄) is a neighborhood of a. We can say even more about open balls.
πŸ”—

Activity 7.2.

πŸ”—
Let (X,d) be a metric space, let a∈X, and let δ>0. In this activity we ask the question, is B(a,δ) a neighborhood of each of its points?
πŸ”—

(a)

πŸ”—
Let b∈B(a,δ). What do we have to do to show that B(a,δ) is a neighborhood of b?
πŸ”—

(b)

πŸ”—
Use Figure 7.4 to help show that B(a,Ξ΄) is a neighborhood of b.
Figure 7.4. B(a,Ξ΄) as a neighborhood of b.
πŸ”—

(c)

πŸ”—
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.
<Prev^TopNext>