Why is $\mathbb{R}^2$ without the x-axis simply connected? So in an Exam Question I found on the Internet, i saw that $\mathbb{R}^2$ without the x-axis is simply connected. But in my opinion, this shouldn't even be connected and thus not simply connected. I think of it as $\mathbb{R}\setminus \{0\}$ which is also not connected. So why should $\mathbb{R}^2$ without the x-axis be connected?
 A: Well, there can be many definitions of simple connection: in this case it is clear that if you consider the definition that involves also the connection of the space, the property is false due to the fact that $\mathbb{R}^2\setminus \{y=0\}$ is not connected. 
You can however define it by: every connected component is simply connected in the previous sense ($\pi_1=0$). In this case both connected components are homeomorphic to $\mathbb{R}^2$, which is simply connected, because it is homeotopically equivalent to a single point.
A: The definition given in the exam (the link to which you provided in the comments) says that a set is simply connected if and only if every loop is homotopic to a constant path ("Ein Bereich $D$ heißt einfach zusammenhängend, wenn sich jeder geschlossene Weg in $D$ stetig auf einen Punkt zusammenziehen lässt."). Well, this is indeed true for $\mathbb R^2$ without the $x$-axis.
This may not be the most common definition (since it does not demand connectedness), but since it's the one used in the exam, the statement is indeed true in that context.
A: A subset $S$ of a topological space $X$ is simply connected if it is path connected and has trivial fundamental group.
Note that the set $\Bbb R^2 \setminus \{(x,y) : y=0 \}$ is not even path connected.
