Question 12: Topology - Compactness in Metric Spaces Instructions: Use data from the link provided below and make sure to give your original work. Plagiarism will not be accepted. You can also use different colors and notations to make your work clearer and more visually appealing. Problem Statement: Prove that in a metric space, every compact set is complete and totally bounded. Theoretical Parts: 1. Definition of Compactness in Metric Spaces: Define compactness within the context of metric spaces and discuss its equivalent characterizations. 2. Complete and Totally Bounded Sets: Define what it means for a set to be complete and totally bounded in a metric space, and explain their significance. 3. Proof: Using the definitions, prove that every compact set in a metric space is both complete and totally bounded. Data Link: https://drive.google.com/drive/folders/1A2bC3dEfGhljKIMnOpQrStUvWxYz1234

Analysis. Show that a set A in ℝ^ 2 is open in the Euclidean metric ⇔ it is open in the max metric.

Question 4: Real Analysis - Compactness Instructions: Use data from the link provided below and make sure to give your original work. Plagiarism will not be accepted. You can also use different colors and notations to make your work clearer and more visually appealing. Problem Statement: Prove that a subset of R is compact if and only if it is closed and bounded. Theoretical Parts: 1. Compactness Definition: Define compactness in the context of Euclidean space and provide key properties. 2. Closed and Bounded Sets: Define closed and bounded sets and explain their significance in real analysis. 3. Proof: Prove the equivalence of compactness, closedness, and boundedness in Rn. Data Link: https://drive.google.com/drive/folders/1hAq6ygz5WZ8P09-F00C_GW_Qm0bf6bGi

How does the principle of duality apply to the study of topological spaces?