What the title said. In a slightly more leisurely fashion:-



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> Let $X$ be a compact, connected subset of $\mathbb{R}^2$ with more than one point, and let $x\in X$. Can $X\smallsetminus\{x\}$ be totally disconnected?
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Note that the [Knaster-Kuratowski fan](http://en.wikipedia.org/wiki/Knaster%E2%80%93Kuratowski_fan) shows that, in the absence of the compactness hypothesis, the answer can be 'yes'.


To give credit where it's due, this question was inspired by one that I was asked by Barry Simon.



Can a connected planar compactum minus a point be totally disconnected?
Being planar has nothing to do with the problem. Suppose a totally disconnects $X$ and choose $b$ different from $a$. By passing to a sub continuum, assume that no proper sub continuum contains both $a$ and $b$. Take non empty disjoint open sets $U$ and $V$ whose union is $ X\sim a$. WLOG $b$ is in $U$, and observe that $U\cup \{a\}$ is closed and connected. 


