Embedding irreducible connected sets

Abstract: 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.