Definition:Underlying Set

Definition

Abstract Algebra

Let $\struct {S, \circ}$ be an algebraic structure.

Then the underlying set of $\struct {S, \circ}$ is the set $S$.

Relational Structure

Let $\left({S, \mathcal R}\right)$ be a relational structure.

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

Metric Space

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

Then the underlying set of $M$ is the set $A$.

Topological Space

Let $T = \left({S, \tau}\right)$ be a topological space.

Then the underlying set of $T$ is the set $S$.