A neighborhood is the area immediately surrounding a point. A neighborhood is usually visualized as a disk with the point at the center. The concept of neighborhood is import in modern mathematics. It has application in geometry, algebra, set theory and calculus. In geometry, the concept of boundary is defined in terms of neighborhood.
Every point has an infinite number of neighborhoods. To see why, imagine a disk with the point at the center. Because points in a plane are continuous, there is another point halfway between the edge and the center of the disk. Use the halfway point to defined another neighborhood. The new neighborhood has a radius of half the original neighborhood. Now imagine a point halfway between the edge and the center of the new disk. This can be repeated infinitely, since there at least one point between any two other points.
The concept of neighborhood is used to define the boundary of a geometric figure, and to define a continuous function. A boundary point of a geometric figure is a point such that any neighborhood of the point contains points in the figure and points not in the figure. Here is how this works.
Take any interior point of a finite geometric figure. If a neighborhood of an interior point is large enough, it will contain points in the figure and points not in the figure. However, as one takes smaller and smaller neighborhoods, eventually a neighborhood is found that does not include exterior points. The same process can be used to show that any exterior point is not a boundary point. Contrast this with a boundary point.
A boundary is a curve such that all points on one side of the boundary are exterior points, and all points on the other side of the boundary are interior points. Select any point on the boundary. No matter how large or small a neighborhood of a boundary point is selected, the neighborhood will always have at least one point in the interior, and at least one point in the exterior.
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