Singapore University of Social Sciences

Topology of Euclidean Spaces (MTH401)

Applications Open: To be confirmed

Applications Close: To be confirmed

Next Available Intake: To be confirmed

Course Types: To be confirmed

Language: English

Duration: 6 months

Fees: To be confirmed

Area of Interest: Science & Technology

Schemes: To be confirmed

Funding: To be confirmed


Synopsis

MTH401 is an Introduction to analysis in a slightly more abstract setting than first encountered in the 1-variable analysis courses. Fundamental concepts from the 1-variable case are abstracted and generalized. Basic topological properties of Euclidean spaces are introduced. These provide the framework for tackling difficult problems in advanced analysis.

Level: 4
Credit Units: 5
Presentation Pattern: Every July

Topics

  • Norm, distance and inner products in euclidean space; metric spaces.
  • Open sets and closed sets, interior of a set.
  • Accumulation points. Closure of a set, boundary of a set. Sequences in metric spaces.
  • Completeness. Series of vectors.
  • Compactness, Totally bounded sets. Bolzano-Weierstrass Theorem. Heine-Borel Theorem.
  • Nested set property. Path-connected sets. Connected sets.
  • Continuity. Images of compact and connected sets. Operations on continuous mappings.
  • Boundedness of continuous functions on compact sets.
  • Extreme Value Theorem. Intermediate Value Theorem. Uniform continuity.
  • Pointwise and uniform convergence.
  • Cauchy criterion. Weierstrass M-Test.
  • Integration and differentiation of series.

Learning Outcome

  • infer subsets of a given set with specific topological properties;
  • compare topological properties of various subsets of Euclidean space;
  • formulate formal notation correctly and in connection with precise statements in English;
  • verify the validity of various theorems relating continuity of maps and topology of subsets;
  • verify topological properties of given subsets of Euclidean space;
  • verify convergence properties of given sequences and series in normed or metric spaces;
  • create examples of maps and subsets satisfying various topological properties in relation to continuity, convergence or completeness;
  • construct proofs regarding topological properties of given subsets, possibly in relation to continuity, convergence or completeness;
  • create proofs of various properties, identities and relations regarding normed, metric, inner product or topological structures;
  • construct proofs, examples or counter-examples regarding continuity, uniform continuity, convergence or uniform convergence of maps between metric or normed spaces
  • solve a number of problem sets within strict deadlines;
  • examine a range of mathematical techniques to solve a variety of formal problems;
  • formulate solutions to problems individually and/or as part of a team.
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