# 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: