TITLE: Prove that $W \cup S^1$ is connected in the subspace topology of $\mathbb{R^2}$
QUESTION [5 upvotes]: I want to solve the following question:

Prove that the union of $W$ and the unit circle $S^1$ is connected in the subspace topology of $\mathbb{R^2}$ where $W=\{(x, y) \in \mathbb{R^2} | x=(1-e^{-t})cost, y=(1-e^{-t})sint, t \geq0\}$

I know that a connected topological space, $X$ is connected if it does not split into open disjoint non-empty subsets.
Also, for a space $(Y, \tau)$ the subspace topology on $X \subset Y$ is $\tau|_X=\{U \cap X : U \in \tau\}$. 
How can the union be connected, since you are taking the union of non-empty subsets?

REPLY [2 votes]: $W$ is connected (since it is a path). Note that $\overline{W}$ is connected, since it is the closure of a connected set. But $\overline{W}=W \cup S^1$.