Difference between revisions of "MAT5243"

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==Topics List==
 
==Topics List==
 
{| class="wikitable sortable"
 
{| class="wikitable sortable"
! Week !! Topic !! Sections from Tucker's book !! Subtopics !! Prerequisite
+
! Week !! Topic !! Sections from Willard's book !! Subtopics !! Prerequisite
 
|-
 
|-
 
|  1-2   
 
|  1-2   
Line 17: Line 17:
 
|| Real Analysis, undergraduate topology.
 
|| Real Analysis, undergraduate topology.
 
|-
 
|-
|  3-4    
+
|  3   
|| [[Product spaces]]
+
|| [[Topological spaces]]
 
|| 3-5
 
|| 3-5
|| Products and quotient topologies
+
||  
 +
* Neighborhoods
 +
* Bases
 +
* Subbases
 +
|-
 +
|  4-5 
 +
|| [[New spaces from old]]
 +
|| 6-9
 +
||
 +
* Subspaces
 +
* Continuous functions
 +
* Product topologies
 +
* Quotient topologies
 +
|-
 +
|  6-7 
 +
|| [[Convergence]]
 +
|| 10-12
 +
||
 +
* Filters and ultrafilter limits
 +
* Characterization of topologies and continuity through convergence
 
|-
 
|-
 
|  5-7   
 
|  5-7   
 
|| [[Separation and countability]]
 
|| [[Separation and countability]]
|| 6-9
+
|| 13-16
|| Separation and countability axioms, regularity and complete regularity
+
||  
 +
* Separation axioma
 +
* Countability axioms
 +
 
 
|-
 
|-
|  8-10    
+
|  8-9    
 
|| [[Compactness]]
 
|| [[Compactness]]
|| 10-12
+
|| 17-19
|| Characterizations of compactness, connections between compactness and continuity
+
||  
 +
* Characterizations of compactness
 +
* Connections between compactness and continuity
 
|-
 
|-
9-10    
+
10-end    
 
|| [[Metrization]]  
 
|| [[Metrization]]  
|| metrizable spaces
+
|| 35-38
|| Search trees, spanning trees, the Traveling Salesman Problem
+
||  
 +
* Uniform spaces
 +
* Metrization of uniform spaces.
 
|}
 
|}

Latest revision as of 21:56, 25 March 2023

Introduction to the general theory of topological spaces

Sample textbook:

[1] Stephen Willard, General Topology, Dover, 20.04


Topics List

Week Topic Sections from Willard's book Subtopics Prerequisite
1-2 Review of set theory and metric spaces 1-2 Cardinal and ordinal arithmetic, metric and pseudometric spaces, metric topologies. Real Analysis, undergraduate topology.
3 Topological spaces 3-5
  • Neighborhoods
  • Bases
  • Subbases
4-5 New spaces from old 6-9
  • Subspaces
  • Continuous functions
  • Product topologies
  • Quotient topologies
6-7 Convergence 10-12
  • Filters and ultrafilter limits
  • Characterization of topologies and continuity through convergence
5-7 Separation and countability 13-16
  • Separation axioma
  • Countability axioms
8-9 Compactness 17-19
  • Characterizations of compactness
  • Connections between compactness and continuity
10-end Metrization 35-38
  • Uniform spaces
  • Metrization of uniform spaces.