  We show that every connected set $X$ which is irreducible between two points
$a$ and $b$ embeds into the Hilbert cube in a way that $X\cup \{c\}$ is
irreducible between $a$ and $b$ for every point $c$ in the closure of $X$.
Also, a connected set $X$ is indecomposable if and only if for every compactum
$Y\supseteq X$ and $a\in X$ there are two points $b$ and $c$ in the closure of
$X$ such that $X\cup \{b,c\}$ is irreducible between every two points from
$\{a,b,c\}$. Following the proofs of these theorems, we illustrate a cube
embedding of the main example from "On indecomposability of $\beta X$". We
prove the example embeds into the plane.
