Multiple choice question from general topology Let $X =\mathbb{N}\times \mathbb{Q}$ with the subspace topology of $\mathbb{R}^2$ and $P = \{(n,  \frac{1}{n}):  n\in \mathbb{N}\}$ .
Then in the space $X$
Pick out the true statements
1 $P$ is closed but not open
2 $P$ is open but not closed
3 $P$ is both open and closed
4 $P$ is neither open nor closed
what can we say about boundary of $P$ in $X$?
I always struggle to figure out subspace topology. Though i am aware of basic definition and theory of subspace topology. I need a bit explanation here about how to find out subspace topology of $P$? 
Thanks for care

For example, 1 is true. You can see that $P$ is not open by looking at an $\varepsilon$-ball around any point $p = (n, \frac1n )$ in $P$. Then there will be a rational $q$ such that $(n,q)$ is inside the ball hence $P$ is not open. (because $\mathbb Q$ is dense in $\mathbb R$)
Also, it's closed: think about why its complement is open. (You can make an $\varepsilon$-ball around the point that doesn't intersect with $P$)
Now that we have established that 1 is true, we know that 2, 3 and 4 are false.
