This book covers on Compactness on Metric and Topological spaces. Metric spaces and Topological spaces concepts were looked into in depth. The topological properties of metric spaces were also discussed. The major theorems discussed were Tychonoff theorem and also Stone-Chec compactfication theorem. The other theorems covered are the Hausdorff and Weak topologies theorems. Compactness on metric and topological spaces were also covered. The book has worked examples of typical exercises on metric, topological spaces and compactness. The use of propositions, lemmas and collorary were used to support the concept of compactness on metric and topological spaces.