For a topological space $(X,\tau)$, the topology $\tau$ on the set $X$ is a family of subsets called open sets, if $X$, $\emptyset$, any union of the subsets, and any finite intersection of the subsets are in $\tau$. This could be a definition of the term "open sets" with respect to topology. 

For a metric space $(X,d)$, however, we have another definition of open sets, say, a subset $S \subset X$ is an open set if $\forall x \in S$, $\exists \epsilon > 0$ s.t. $B\_{\epsilon}(x):=\{y \in X \;|\; d(x,y) < \epsilon\} \subset S$. This should be identical to the first definition in terms of metric topology. **But how to show the second definition as a reduced version of the first one?** 

In addition, the concept of open sets seems strongly related to the concept of continuous functions. A generalized definition of the continuity in topological space is the inverse image of every open set is open. Traditionally, on the other hand, one defines the uniform continuity of a function $f(x)$ by $\forall x\_1, x\_2 \in X$, $\exists \delta > 0$ s.t. $\|f(x\_1) - f(x\_2)\| < \epsilon$ for $\forall \epsilon$ if $\|x\_1 - x\_2\| < \delta$. **How to find the connections between the definitions in analysis and in topology?**