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

Elaborate on the role of duality in topological spaces.

25 Topology: Properties of Compact Spaces Task: Refer to Question 25 in the provided document. Link: https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC44OAZF/view?usp=sharing

I need help coming up with examples and maybe an explanation. The subject is Real Analysis.

Q/Discuss the following statement: "A complete metrice space metrice space need not be compact Give an example ? 41

Show that every metric space is an Hausdorff space.

I have the following question:  Let A be a set and d : A × A → {0, 1, 2} a symmetric setfunction that satisfies d(x, y) = 0 ⇔ x = y. Show that (A, d) is a metricspace. please if able write the proof in a detailed manner with some explanation. I seem to struggle with proving the obvious or proving something the wrong way like for example with the triangle inequality somehow it takes way to long to prove. Thank you in advance.

Please solve i,ii,iii,iv

Question 2. Show that the set of all real numbers constitutes an incomplete metric space if we choose d(x, y) = |tanr – tan'y. Question 3 . If X and Y are isometric and X is complete, show that Y is complete. fx 6 Gold أدوات تأثیرات تجميل لاصق

Instructions to follow: * Give original work *Support your work with examples and graphs where required * Follow The references: Kreyszig, Rudin and Robert. G. Bartle. Reference Books: C.D. Aliprantis and O. Burkinshaw, Principles of Real Analysis, 3rd Edition, Harcourt Asia, (2000) Fixed Point Theorem in Banach Spaces Question: Prove the Banach Fixed Point Theorem (Contraction Mapping Theorem) for a complete metric space X and a contraction mapping T : XX. Discuss applications of this theorem in solving integral and differential equations, and present an example where this theorem is crucial. J. Bak and D.J. Newman, Complex Analysis, 2nd Edition, Springer Indian Reprint, (2009) Sobolev Spaces and Embedding Theorems Bartle and Sherbert, Introductory Real Analysis, 3rd edition, Wiley International, (2001) E. Kreyszig, Introductory Functional Analysis with Applications, Wiley Singapore Edition, (2001). S. Kumaresun, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis…