Definition:Subcontinuum

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

Let $H \subseteq S$ be a continuum in $T$.

Let $K \subseteq H$ also be a continuum in $T$.

Then $K$ is a subcontinuum (plural: subcontinua) of $H$.

Proper Subcontinuum
Let $K$ be a subcontinuum of $H$ such that $K \ne H$.

That is, let $K$ be a proper subset of $H$.

Then $K$ is a proper subcontinuum of $H$.