Definition:Infimum

Ordered Set
Let $$\left({S; \preceq}\right)$$ be a poset.

Let $$T \subseteq S$$.

An element $$c \in S$$ is the infimum of $$T$$ in $$S$$ if:


 * 1) $$c$$ is a lower bound of $$T$$ in $$S$$;
 * 2) $$d \preceq c$$ for all lower bounds $$d$$ of $$T$$ in $$S$$.

Plural: Infima.

The infimum of $$T$$ is denoted $$\inf \left({T}\right)$$.

The infimum of $$x_1, x_2, \ldots, x_n$$ is denoted $$\inf \left\{{x_1, x_2, \ldots, x_n}\right\}$$.

If there exists an infimum of $$T$$ (in $$S$$), we say that $$T$$ admits an infimum (in $$S$$).

The infimum of $$T$$ is often called the greatest lower bound of $$T$$ and denoted $$\operatorname{glb} \left({T}\right)$$.

Mapping
Let $$f$$ be a mapping defined on a poset $$\left({S; \preceq}\right)$$.

Let $$f$$ be bounded below on $$S$$.

It follows from the Continuum Property that the range of $$f$$ has an infimum on $$S$$.

Thus $$\inf_{x \in S} f \left({x}\right) = \inf f \left({S}\right)$$.