= Rational comb with a deleted attachment point
{title2=$C=(\mathbb Q\times[0,1])\cup(\mathbb R\times\{1\})$}
The displayed subspace of the <Euclidean plane> is <path-connected>: each vertical rational tooth joins the horizontal line at height one. Delete $(r,1)$ for rational $r$. The residual tooth $V=\{r\}\times[0,1)$ cannot be joined by a <path> to the top line. Before a path first reaches height one, its first coordinate is a continuous rational-valued function on an <interval> and must therefore be constant; it would reach the deleted point.
The punctured space remains <connected>. Its portions $L,R$ on either side of $x=r$ are <path-connected>, and $V\subseteq\overline L\cap\overline R$ by the <density of the rational numbers>. Adjoining <limit points> to a <connected set> preserves connectedness: a separation would place the connected subset on one side and prevent the other side from containing its limit points. Hence $L\cup V$ and $R\cup V$ are connected and intersect in $V$. Their union is connected. This example distinguishes <connectedness> from <path-connected> using the loss of a single attachment point.
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