Exercise 19, § 4

A compact connected space $X$ is said to be irreducible between two of its points $x, y$ if there is no compact connected subset of $X$ other than $X$ itself which contains both $x$ and $y$.

a) If $x$ and $y$ are two distinct points of a compact connected space $X$, show that $X$ has a compact connected subspace $K$ which is irreducible between $x$ and $y$ (use Exercise 14 and Zorn’s Lemma).

b) Show that a compact connected space having at least two points cannot be irreducible between every pair of its points (use Exercise 17).

c) Let $X$ be a compact connected space, and suppose that there is a point $a \in X$ which has a connected closed neighbourhood $V \neq X$. If $x$ and $y$ are any two points of $C_V$, show that there is a compact connected subset of $X$ which contains $x$ and one of the points $a, y$ but not the other (use Exercise 17). Under the same hypotheses, show that if $x, y, z$ are any three distinct points of $X$, then $X$ cannot be simultaneously irreducible between $x$ and $y$, between $y$ and $z$, and between $x$ and $z$.