  We show that if $X$ is a separable locally compact Hausdorff connected space
with fewer than $\mathfrak c$ non-cut points, then $X$ embeds into a dendrite
$D\subseteq \mathbb R ^2$, and the set of non-cut points of $X$ is a nowhere
dense $G_\delta$-set. We then prove a Tychonoff cut-point space $X$ is weakly
orderable if and only if $\beta X$ is an irreducible continuum. Finally, we
show every separable metrizable cut-point space densely embeds into a reducible
continuum with no cut points. By contrast, there is a Tychonoff cut-point space
each of whose compactifications has the same cut point. The example raises some
questions about persistent cut points in Tychonoff spaces.
