Show that $A∪B$ is connected

Question

Define subsets $A,B$ of $\Bbb{R}^2$ by

$$A=\{(x,y)∈\Bbb{R}^2 : x^2+y^2=1\}$$ $$B=\{(a(t) \cos t, a(t) \sin t ) \in \Bbb{R}^2 : 1≤t<∞\}$$ where $a(t)=\frac{1-t}{t}$

Show that $X=A∪B$ is a connected set.

This problem is about the connectedness in general topology.

I don't know how to approach.

Thank you in advance!

Answer 1

The set $B$ is connected, since it is the image of $[1,+\infty)$ by a continuous function. Therefore, $\overline B$ is connected too. But $\overline B=A\cup B$.