MAT4273

From Department of Mathematics at UTSA
Revision as of 20:54, 13 August 2020 by James.kercheville (talk | contribs) (Added content to the table (weeks 5-7))
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The textbook(s) for this course is [1] Undergraduate Topology, by R. H. Kasriel; topics from [2] Introduction to Analysis by W. R. Wade (Stone-Weierstrass Theorem).

A comprehensive list of all undergraduate math courses at UTSA can be found here.

The Wikipedia summary of topology and its history.

Topics List

Date Sections Topics Prerequisite Skills Student Learning Outcomes
Week 1
30-31

Algebraic Structure of the Real Numbers

  • Review of algebraic properties of the Real Numbers


Week 1
30-31


Distance Between Two Points in the Real Numbers

  • The distance between two point on the real line
Week 1
30-31

Limit of a sequence in the Real Numbers

  • The definition of a limit of a sequence


Week 1
30-31

The Nested Interval Theorem for the Real Numbers

  • The nested intervals property in the real numbers


Week 2
34-38


Cauchy-Schwarz Formula

  • The Cauchy-Schwarz Formula


Week 2
34-38

Distance Between Two Points in Higher Dimensions

  • Distance formula for higher dimensions


Week 2
34-38

Open Subsets in Higher Dimensions

  • Definition of an open set
  • Definition of an interior point


Week 2
34-38

Limit Points (or Cluster Points) in Higher Dimensions

  • Definition of a limit point (or cluster point)
  • The limit point as the limit of a sequence


Week 3
39-40

Closed Subsets in Higher Dimensions

  • Definition of a closed set
  • Properties of a closed set


Week 3
39-40

Bounded sets in Higher Dimensions

  • Definition of a bounded set
  • Properties of a bounded set


Week 3
39-40

The Nested Interval Theorem in Higher Dimensions

  • The nested intervals theorem in higher dimensions


Week 3
39-40

The Bolzano-Weierstrass Theorem in Higher Dimensions

  • The Bolzano Weierstrass Theorem in higher dimensions


Week 4
41-43

Convergent Sequences and the Cauchy Criterion in Higher Dimension

  • Definition of a convergent sequence in higher dimensions
  • The Cauchy criterion for convergence of sequences in higher dimensions


Week 4
41-43


Heine-Borel Theorem

  • Definition of an open cover for a set
  • Definition of compactness
  • The Heine-Borel Theorem


Week 4
41-43


Lindeloff Theorem

  • The Lindeloff Theorem


Week 5
45 and 75-76

Topological Spaces

  • Definition of a Topological space
  • Examples of Topological spaces
  • Basic theorems concerning topological spaces


Week 5
45 and 75-76


Distance Functions, Metrics

  • Criteria for a function to be a distance function
  • Common distance functions


Week 5
45 and 75-76

Metric Spaces

  • The definition of a metric space
  • Basic examples of metric spaces


Week 6
46-47

Open Sets and Closed Sets in Metric Spaces

  • Definition of an open set in a metric space
  • Definition of a closed set in a metric space
  • Basic examples of open and closed sets in metric spaces


Week 7
48-49

A Topology Given By A Metric

  • Typologies given by metrics


Week 7
48-49

Subspaces of Metric Spaces

  • Subspace topologies in metric spaces


Week 5
41-43


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Week 5
41-43


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Week 5
41-43


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