MST-003 IGNOU Solved Assignment 2026-27

MST-003 Solved Assignment: Comprehensive TMA Coverage

The MST-003 Solved Assignment addresses all mandatory components of the course, including Real Analysis, Metric Spaces, Topological Spaces, Compactness and Connectedness, and Functional Analysis Basics—directly referenced from the official IGNOU CBCS syllabus. Below is a structured breakdown of the solved TMA with verified answers.

Key Syllabus Units Covered

• Unit 1: Real Analysis Foundations: Solutions for problems on sequences, series convergence, and Cauchy sequences, with step-by-step proofs adhering to ε-δ definitions.
• Unit 2: Metric Spaces: Detailed explanations of metric space axioms, completeness, and examples (e.g., p-norms in ℝⁿ), including proofs for compactness in metric spaces.
• Unit 3: Topological Spaces: Coverage of open sets, closure, interior, and boundary, with explicit reference to the Tychonoff theorem and its implications.
• Unit 4: Compactness and Connectedness: Solutions for problems on compact subsets, Heine-Borel theorem, and connectedness criteria (e.g., path-connected vs. locally connected).
• Unit 5: Functional Analysis Basics: Introduction to normed spaces, Banach spaces, and Hilbert spaces, with solved examples on completeness and orthonormal bases.

Assignment Structure & Solutions

The TMA is divided into three sections with prescribed word limits (500/250/100 words), each solved with:

• Section A (500 words): Comprehensive answers for 10-marks questions on real analysis theorems (e.g., Bolzano-Weierstrass, Intermediate Value Theorem) with rigorous proofs.
• Section B (250 words): Short-answer solutions for 5-marks questions on metric space properties, including counterexamples for non-complete spaces.
• Section C (100 words): Precise definitions and explanations for 3-marks questions on topological concepts (e.g., basis of a topology, quotient topology).

Subject-Specific FAQs

• Q: How do I distinguish between sequential compactness and compactness in metric spaces?

  A: Sequential compactness requires every sequence to have a convergent subsequence, while compactness in metric spaces (per Heine-Borel) requires every open cover to have a finite subcover. The two coincide in metric spaces but differ in general topological spaces.

• Q: Are there common mistakes students make in proving a space is complete?

  A: Yes—students often overlook verifying the Cauchy criterion