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MTH634 : Topology

Course Overview

Course Synopsis

This course will introduce students to essential notions in topology, such as topological spaces, continuous functions, and compactness.

Course Learning Outcomes

At the end of the course, students will be able to:

  • Students should get an introduction to the field of topology, with awareness of the aspects of the subject that are basics to higher mathematics.
  • Students should learn the fundamentals of topology and topological spaces including open set, closed set, limit point, continuous maps, Homeomorphism, separation axioms, connectedness locally connectedness, path connectedness and compactness.
  • Students are expected to learn how to write mathematical proofs in logical manner by using important theorems and properties of topological spaces.
  • Students are expected not only to grasp the concepts of topology and apply them, but also to continue with their overall mathematical development.

Course Calendar

1 Introduction to Topology
2 Topology on a Set:1
3 Topology on a Set:2
4 Examples of Topologies on a Set:1
5 Examples of Topologies on a Set
6 Examples of Topologies on a Set2
7 More examples
8 Some special topologies
9 Cofinite topology
10 Topological space
11 Arbitrary Intersection
12 Open and Closed sets

13 Properties of clooection of closed sets
14 Intersection of Topologies
15 Union of topologies
16 Usual topology on R
17 Usual topology on R^2
18 Usual topology on R^n
19 Comparison of topologies
20 Limit Point of a Set

21 Derived Set
22 Derived Set: Special Topologies
23 Derived Set: Some Results
24 Closed set and derived set
25 Closed set and derived set 2
26 Criterian for Openness
27 Closure of Set
28 Computation of Closure
29 Properties of a Closure of Set

30 Dense Set
31 Convergent Sequence
32 Convergent Sequence: Examples
33 Interior point
34 Interior of a Set
35 Properties of Interior
Assignment 1
Quiz 1
36 Exterior of a Set
37 Boundary of a Set
38 Partition of a Set

39 Neighborhood system
40 Neighborhood system: Properties
41 Subspace topology
42 Subspace topology: Proof
43 Relative Open Sets
44 Basis For a Topology
45 Basis For a Topology: Examples
46 Criteria For a Collection to be a Basis
47 Basis for Some Topologies
48 Basis for Usual Topology R^n
49 Topologies on Real Line
50 Basis For Subspace

51 Basis: Comparison of Topologies
52 Finness of R_l
53 Finness of R_k
54 Subasis
55 Topology Generated by Collection
56 Topology Generated by Collection: Examples
57 Topology Generated by Collection on R: Examples
58 Subbasis for Subspace
59 Smallest Topology Containing Topologies
60 Largest Topology Contained in Topologies
61 Local Base
62 Example: Local Base
63 Continuous Function
64 Example:Continuous Function
65 Constant Function
66 Function into Indiscrete Space
67 Function into discrete Space

68 Continuity and Open Sets
69 Continuity and Open Sets: examples
70 Continuity and Comparability
71 Continuity and Basis
72 Continuity and Subbasis
Quiz 2
73 Absolute value Function
74 Composition of Continuous Functions
75 Continous Funtion: Another Definition
76 Restriction of Function
77 Continuous Function and Arbitrary Closeness
78 Continuity at a Point
79 Continuity at a Point: examples
80 Continuity at Ever Point and Continuous Function

81 Topology Induced By Function
82 Topology Induced By Function: Examples
83 Topology on Real Line and Linear Function
84 Open Mapping
85 Open Mapping and Basis
Mid semester exam
86 Closed mapping
87 Projection mapping
88 Continuity and open mapping
89 Openness and restriction mapping
90 Homomorphism and honomorphic spaces
91 Examples: Homomorphism
92 Bicontinuity and open mapping
93 Bicontinuity and bijection
94 Homomorphism and equivalence relation
95 Topological properties
96 Metric on a set

97 Examples of metric
98 More Examples on Metric
99 Usual metric on R^n
100 Unusual metric on R^n
101 Open Ball
102 Open ball examples
103 Open ball as open set
104 Open Balls centered at one point
105 Properties of open balls
106 Metric topology
107 Examples: Metric Topology
108 Metrizable Spaces
109 Examples: Metrizable spaces
110 Examples: No-metrizable spaces
111 First countable spaces
112 Examples: First countable space

113 Second countable space
114 Examples: Second countable space
115 Second contable implies first countable
116 First countable does not imply second
117 Open cover
118 Example: Open cover
119 Open subcover
120 Lindelof space
121 Second countable and Lindelof space
122 Separable spaces
123 Examples: Separable spaces
124 More examples Seaparable spaces
125 Subspace of separable space
Quiz 3

126 Metric apce and separability
127 Seaparation Axiom
128 T_0 space
129 Examples: T_0 space
130 T_1 space
131 Examples: T_1 space
132 T_0 does not imply T-1
133 Criteria for T_1 space
134 Subspace of T_1 space
135 Properties of T_1 space
136 Properties of T-1 space: II
137 T-2/Hausdorff spaces
138 Examples: hausdorff spaces

139 Subspace of Hausdorff spaces
140 Metric spaces are Hausdorff spaces
141 T1 does not imply T2
142 Unique Limit point
143 Unique limit point theorem
144 Regular Spaces
145 Example: Regular Spaces
146 Normal Space
147 Examples: Normal spaces
148 Metric spaces are normal
149 T_i properties: An overview
150 Urysohn's Lemma
Quiz 4

151 Compact Spaces
152 Examples: Compact space
153 Open intervals in R
154 Cofinite topology and compactness
155 Compactness and open collection
156 Closed subspace of compact space
157 Compact subspace of Hausdorff space
158 Compact space under continuous map
159 Compactness and Homomorphism
160 Application: Compactness and Homomorphism
161 Finite union of Compact spaces
162 Compact Hausdorff space is normal
163 Separated sets

164 Examples:Separated sets
165 Connected spaces
166 Examples: Connected spaces
167 Connected spaces and open closed sets
168 Connected sets and its closure
169 Image of connected space
170 Application: Image of connected set
171 Connected subsets of real line
172 Connectedness and fixed point theorem

173 Union of connected subsets
174 Connected component
175 Examples:Connected component
176 Connected component is closed
177 Connected component is not open
178 Locally connected spaces
179 Examples: Locally connected spaces
180 Connectedness and local Connectedness
181 Path connected spaces
182 Facts: Path connected spaces 1
183 Facts: Path connected spaces 2

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