Definition:Subcontinuum
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Definition
Let $T = \struct {S, \tau}$ 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 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 $H$.
Linguistic Note
The plural of subcontinuum is subcontinua, although the technically-erroneous form subcontinuums can occasionally be found.