Abstract: We are going to discuss some standard problems regarding uniform approximation in the complex domain. In particular, we are going to focus our attention on Arakelian's theorem, i.e., uniform holomorphic approximation on closed sets. These topics have been extensively studied in the past since the celebrated theorem by Mergelyan appeared in the early 50's. What we are going to address in this talk is perhaps a not so noted topological viewpoint of the subject. More precisely, we will motivate the following characterization of Arakelian sets and examine applications: "A closed set $F\subset \mathbb{C}$ is an Arakelian set in the plane, if and only if it has a base of simply connected open neighborhoods."