Irreducible continuum

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Let $X$ be a continuum. We say that $X$ is irreducible (between $p$ and $q$) if there exist $p,q \in X$ such that no proper subcontinuum of $X$ contains $\{p,q\}$. We call both the points $p$ and $q$ in the definition of irreducible to be points of irreducibility. We use the notation $\kappa(p)$ to denote the composant of $X$ containing $p$.

Properties

Theorem: If $X$ is a nondegenerate continuum and $p \in X$, then $p$ is a point if irreducibility of $X$ if and only if the composant containing $p$ is not $X$ itself.

Proof:

Corollary: A nondegenerate continuum $X$ is irreducible if and only if some composant of $X$ is a proper subset of $X$.

Proof:

Relationship between irreducibility and indecomposability

Theorem: (Kuratowski's Theorem) Let $X$ be a continuum and $p \in X$. Then $p$ is a point of irreducibility of $X$ if and only if $X$ is not the union of two proper subcontinua both of which contain $p$.

Proof:

Sorgenfrey's theorem on triods and irreducibility