How can a family of spherical neighborhood to be a topology of a metric space?

Question

I am learning topology. And my textbook states that for a metric space $X$,$\{U|U = \cup_{\alpha}B_{\alpha}(x,\epsilon_{x})\}$ can be a topology of that. My question is that if $X = \{a_1,a_2\}$, how can $\{U|U = \cup_{\alpha}B_{\alpha}(x,\epsilon_{x})\}$ be its topology?

Answer 1

Just think about the $\epsilon$-ball $B(a_1,\epsilon)$ centered at $a_1$.

Since $X=\{a_1,a_2\}$ is a metric space, denote the distance between $a_1$ and $a_2$ by $d(a_1,a_2)>0$.

1. If $0<\epsilon\leq d(a_1,a_2)$, then $B(a_1,\epsilon)=\{x\in X\mid d(a_1,x)<\epsilon\}=\{a_1\}$.

2. If $\epsilon>d(a_1,a_2)$, then $B(a_1,\epsilon)=\{x\in X\mid d(a_1,x)<\epsilon\}=\{a_1,a_2\}=X$.

Similarly, we have $B(a_2,\epsilon)=\{a_2\}$ or $X$ according to $\epsilon$.

Therefore, the topology on $X$ is nothing but the discrete topology, $\{\varnothing, \{a_1\}, \{a_2\}, X\}$.