Metric spaces, distance. Diameter of a metric space. Bounded, unbounded metric spaces. Open, closed balls. Open, closed sets. Union and intersection of open or closed sets. Definition of interior, closure and boundary of a set and their properties. Metric spaces and connected sets. Connected sets in \(\mathbb {R}\) . Polygonal. Open connected sets in \(\mathbb {C}\) . Convergent sequences, limit points. The closure of a set coincides with its limit points. Dense sets. Cauchy sequences. Convergent sequences are Cauchy sequences. A Cauchy sequence that admits a convergent subsequence is convergent. Metric spaces and complete sets. Completeness of \(\mathbb {C}\) (assuming \(\mathbb {R}\) complete). A subset of a complete metric space is complete if and only if it is closed. Metric spaces and (sequentially) compact sets. A compact metric space is complete. Totally bounded metric spaces. A totally bounded metric space is bounded. A metric space is compact if and only if it is complete and totally bounded. A subset of \(\mathbb {R}^n\) is compact if and only if it is closed and bounded.