Triod

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A continuum $X$ is called a triod if there is a subcontinuum $Z$ of $X$ such that $X \setminus Z$ is the union of three nonempty sets each two of which are mutually separated in $X$. $X$ is called a weak triod if $X=X_1 \oplus X_2 \oplus X_3$ and $\bigcap_{i=1}^3 X_i = \emptyset$, where $\oplus$ denotes the essential sum.

Properties

Proposition: Every triod is a weak triod.

Proof:

Theorem: If $X$ is unicoherent, then $X$ is a triod if and only if $X$ is a weak triod.

Proof:

Sorgenfrey's theorem on triods and irreducibility