Definition:Underlying Set

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 $X$.