Definition:Underlying Set

Definition

Abstract Algebra

Let $\left({S, \circ}\right)$ be an algebraic structure.

Then the underlying set of $\left({S, \circ}\right)$ is the set $S$.

Metric Spaces

Let $\left({X, d}\right)$ be a metric space.

Then the underlying set of $\left({X, d}\right)$ is the set $X$.

Topological Spaces

Let $\left({X, d}\right)$ be a topological space.

Then the underlying set of $\left({X, d}\right)$ is the set $X$.

Sources

• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{II}$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{III}$