Connectedness of A Set

Question

Munkres asserts that the following:

$$X = \{x \times y | y = 0\}\cup\{x \times y | x>0 \text{ and }y = 1/x\}$$

is not connected. (Basically to two pieces are the graph $1/x$ and its asymptote at $0$). I'm not seeing this though:

1. If I try to find an open set in $\mathbb{R^2}$ containing $\{x \times y | y = 0\}$ it will also contain the tail of $\{x \times y | x>0 \text{ and }y = 1/x\}$
2. Isn't the limit of each contained in the other? (Munkres claims its not, then stops explaining)

Answer 1

HINT : A non-connected set is the union of two disjoints closed subsets. Try to prove that your two natural subsets are closed.