Is a topological space without cut points connected? I'm trying to prove that having a cut point is a topological invariant, and my proof hinges on this claim that a space without cut points is connected, but I haven't proved this claim yet, so I'm not sure if it is true. I continue to contemplate...

Surely a space without cut points isn't necessarily connected. Take for instance $]0,1[\cup]2,3[\subset\mathbb{R}$ with Euclidean topology. In fact, any not connected space does not have any cut points, since cut points are only defined for connected spaces.

For the actual proof:

Note that if $X$ has a cut point $x$, and $X$ is homeomorphic to some other space $Y$ by the homeomorphism $f$, then since $X$ is connected, $Y$ is connected. Now $X-\\{x\\}$ is homeomorphic to $Y-\\{f(x)\\}$. Since $X-\\{x\\}$ is not connected, $Y-\\{f(x)\\}$ is not connected, hence $f(x)$ is a cut point of $Y$.