Def/Metric space

 =Balls= If $(X,d)$ is a metric space, then we may consider open and closed balls in $X$. If $c \in X$, and $r$ is a positive real number, then we define: $$B_r(c) = \{ x \in X \mbox{ such that } d(x,c) < r \}.$$ $$\bar B_r(c) = \{ x \in X \mbox{ such that } d(x,c) \leq r \} .$$
 * The defines::open ball, with center $c$ and radius $r$, is the subset:
 * The defines::closed ball, with center $c$ and radius $r$, is the subset:

=Open sets= If $(X,d)$ is a metric space, then $(X,d)$ has a natural topology $T_d$ with basis consisting of open balls. Specifically, we declare that a subset $U \subset X$ is defines::open if: $$\forall u \in U, \exists r \in RR, \mbox{ such that } r > 0 \mbox{ and } B_r(u) \subset U.$$ In other words, a subset $U \subset X$ is open if, at every point of $U$, we can find an open ball centered at that point and contained within $U$.

Let $T_d$ be the set of open subsets of $X$, as defined above. The set $T_d$ is sometimes called the defines::metric topology, or the topology induced by the metric $d$. This definition of openness satisfies the axioms for a topological space. Namely, we find the following about open sets:
 * 1)  The empty set is open.
 * 2)  The union of an arbitrary set of open sets is open.
 * 3)  The intersection of a finite number of open sets is open.

Indeed, the empty set is open, since the truth of the sentence "$\forall u \in \emptyset, \exists r \in \RR, \ldots$" is automatic (truth by false premise). Next, we check that the union of open sets is open:

Finally, we check that the finite intersection of open sets is open.