In $\mathbb{R}^3$ with the usual topology, let:
\begin{align*} V &= \{(x, y, z)\in \mathbb{R}^3 : x^2+y^2+z^2 = 1,\,y\neq 0\} \\ W &= \{(x, y, z)\in\mathbb{R}^3 : y = 0\} \end{align*}
I have to check for connectedness and compactness of $V \cup W$.
Here is my approach: Subspace $V$ being closed and bounded and hence compact while $W$ is closed but unbounded and hence not compact. This implies that $V\cup W$ is not compact.
For connectedness I know that union of two non-disjoint connected sets is connected. Intutively I think that both are non intersecting. But I am not sure with this.
Is there any other way to solve this problem?
Thanks for helping me.