I am taking a topology course and we are now learning open and closed set. I am a bit confused to how to prove that a set is closed or opened, how should I approach these kind of problems. For example:
Question 1: Let $(\mathcal{X},d)$ be an arbitrary metric space. Prove that any set which contains a finite number of points $\{x_1,x_2,\ldots,x_n\}$ is closed.
Solution: If we take the point $x_i$ where $1\leq i\leq n$ and no matter how small we make $r$, in the ball some points are outside of our set. Hence the set is closed.
Question 2: Let $(\mathcal{X},d)$ be an arbitrary metric space. Prove that $B(\mathcal{X},r)=\{y\in\mathcal{X}:d(x,y)<r\}$ is open.
Solution: Take a point $y_0$ in the ball, and make a ball with radius $\frac{1}{2}(r-d(x,y_0))$, the all the points in this ball are in the actual ball in the question. Hence the ball in the question is open.
Are these proves right? I myself do not feel that I am proving anything. Could you please teach me the correct proof if these are not correct?
Thanks