General topology, compact sets, neighborhood

Question

I'm really struggling in writing the proof of these statements

My answers are: 1)true 2)true 3)false but i can't supply the proof for this. Any help please? I don't know how to write proofs. I'm new to this course

Answer 1

By metric spaces your answers are correct.

1) If $p\neq q$ then $d(p,q)>0$ and for $\epsilon=\frac12d(p,q)>0$ the sets $U_p=\{x\in X\mid d(p,x)<\epsilon\}$ and $U_q=\{x\in X\mid d(q,x)<\epsilon\}$ are disjoint open sets containing $p$ and $q$ respectively.

2) Similarly for $i=1,\dots,k$ let $U_i$ denote an open set containing $p_i$ and let $V_i$ denote an open set containing $q$ with $U_i\cap V_i=\varnothing$. Now take the union of the $\bigcup_{i=1}^nU_i$ and $V=\bigcap_{i=1}^n V_i$.

3)Counterexample: $\mathbb R$ with usual topology with $p_i=\frac1{i}$ and $q=0$.