Definition:System of Neighborhoods

Definition

Let $M = \struct {A, d}$ be a metric space.

Let $a \in A$.

Let $\NN_a$ be the set of all neighborhoods of $a$ in $M$.

Then $\NN_a$ is the system of neighborhoods of the point $a$.

Also known as

The system of neighborhoods of a point can also be referred to as the complete system of neighborhoods of that point.

Also see

• Results about systems of neighborhoods can be found here.

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 4$: Open Balls and Neighborhoods: Definition $4.4$