Exercise. Consider the open ball $S = \{(x,y) \in \Bbb R^2: (x-1)^2 + y^2 < 1\}$ and the singleton $\{(0,0)\}$. Show that $S \cup \{(0,0)\}$ is connected.
My attempt. I know for a fact that $S$ is connected (just like any other open ball in $\Bbb R^2$) and that singletons are also connected. So initally I thought about using the fact that the union of connected sets is connected but obviously in this case we have that the intersection of the singleton and $S$ is empty and thus the result is not applicable. On the other hand, I also know that $S$ is path-connected (thus connected). What I thought after this was to define a continuous mapping from $[0,1]$ to the union we want and use the fact that the continuous image of a connected set is also connected, but I am having some trouble defining such function.
I can see graphically that this is indeed a connected set, but I want to show it algebrically!
Thanks for any help in advance.
Your set is the union of two connected sets with non-empty intersection: $S$ and the line segment from $(0,0)$ to $(1,0)$. Therefore, it is connected.