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Topology
MATH 342 Spring 2026
Kyle Ormsby
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Front Matter
1
Course Documents
1.1
Syllabus
Course Information
2
Course Notes
2.1
Week 1
Monday
Welcome to the Generous Arena
So what is a topology?
Examples of topologies
Wednesday
Continuity
Topological properties
Building continuous functions
Friday
2.2
Week 2
Monday
Topology from the categorical viewpoint
Categories
Functors
Wednesday
More on functors
New topologies from old
The subspace topology
Friday
The quotient topology
2.3
Week 3
Monday
Real projective space
Pushouts
Wednesday
Pushouts in Top
Categorical products and the product topology
Friday
2.4
Week 4
Monday
Separation of points
Wednesday
Closure and limit points
Sequences and limits
Sequences, closure, and continuity
Friday
A first glimpse of compactness
2.5
Week 5
Monday
Compactness via topological induction
Inheritance of compactness
Wednesday
Separation of disjoint compact subspaces
Friday
2.6
Week 6
Monday
The fundamental theorem of topology
When do quotients separate points?
Wednesday
Connectedness
Paths and path connectedness
Friday
Closed intervals are connected
Path-connected implies connected
Inheritance of connectedness
2.7
Week 7
Monday
Wednesday
Friday
Connected components
Path-components and
\(\pi_0\)
2.8
Week 8
Monday
Comparing connected and path-components
Local connectedness
Wednesday
Natural transformations
Adjunctions
Friday
Units and counits of adjunctions
Adjoints to the forgetful functor from spaces to sets
Stone-Čech compactification
2.9
Week 9
Monday
Wednesday
Motivation: topologies on function spaces
Splitting and conjoining topologies
The compact-open topology
Metrics on function spaces
Friday
The product topology via mapping spaces
Local compactness
Continuity of evaluation and the exponential adjunction
Applications of the exponential adjunction
Limitations and the role of local compactness
Application: Stone-Čech compactification via mapping spaces
3
Homework
3.1
Homework 01
3.2
Homework 02
3.3
Homework 03
3.4
Homework 04
3.5
Homework 05
3.6
Homework 06
3.7
Homework 07
Topology
MATH 342 Spring 2026
Kyle Ormsby
Department of Mathematics & Statistics
Reed College
Last Updated: April 3, 2026
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