Using the definition
$\Omega\subset\Bbb R^n$ is connected iff it cannot be written as the union of two disjoint non empty opens $A$ and $B$ of $\Bbb R^n$,
how would you prove that $\Bbb R^2\backslash\{(0,0)\}$ ?
I know it is "path connected" by the standard definition of path connected, but in the definition that we use that does not imply connected.
The definition you wrote does mean that path connected implies connected. That is, the statement
is true, if we use the definition of connectedness that you wrote.
However, if you want to prove $\mathbb R\setminus\{(0,0)\}$ is connected without resorting to the theorem above, you can still do it by sort of re-proving that theorem.
Assume that $X,Y$ are both closed, open and nonempty subsets of $\mathbb R\setminus\{(0,0)\}$. Then, as they are nonempty, there exists $x\in X, y\in Y$. Now, draw a path from $x$ and $y$, and prove that the path cannot be continuous and exist in $X$ and $Y$ at the same time.