145B Spring 2025 - Discussion

Course Info: Introduction to Topology II

• Office Hours: 

  • Location: Skye 281

  • Time: 11am Thursdays

• Quizzes and Exams:

  • Quiz 1 - 10th April

  • Quiz 2 - 1st May

  • Quiz 3 - 22th May

  • Final Exam - 9th June

Week 1

• Discussion - 3rd April

  • Topics: Review of 145A, basis, product topology, subspace topology.

  • Homework 2

Week 2 Quiz!! 📚

• Discussion - 10th April

  • Topics: Compactness, Sequential Compactness, Heine-Borel

  • Note 1: Compactness implies sequential compactness but not the converse is not true in general. If a topological space is sequentially compact AND is either metric or second countable, then it is also compact. Couple of references: Page 99 of Topological Manifolds by J. M. Lee, or in the special case of R^n, check here.

  • Note 2: In R^n with standard topology, sequential compactness is equivalent to compactness since the standard metric induces the standard topology on R^n. 

  • Example 1: {R} is an open cover of R, and has a finite subcover ({R} itself), does this make R compact?

  • Homework 3

Week 3

• Discussion - 17th April

  • Topics: More on Compactness, Connectedness

  • Note 1: One can construct compact spaces by using continuous maps, product topologies, and quotient topologies. One special example discussed was the one point compactification. Also see page 185 of Munkres.

  • Note 2: Connectedness tells us that the topological space can not be separated by two disjoint open sets. One can consider the "largest" connected subsets of a topological space, these are called connected components. The main theorem of connectedness is that continuous maps preserve connectivity.

  • Homework 4

Week 4

• Discussion - 24th April

  • Topics: More examples on sequential compactness

  • Note: Compactness and Compactification by Terence Tao

  • Homework 5

Week 5 Quiz!! 📚

• Discussion - 1st May 

  • Topics: Quotient topology

  • Note 1: A surjective map from a topological space can be used to put a topology onto the target space. One simple way to construct a surjective map is to identify subsets of your space using an equivalence relation. The quotient topology put on te target space guarantees that the quotient map is continuous.

  • Note 2: We may put a topology subset of linear subspaces on R^n using an equivalence relation on R^n minus the origin. This space is called real projective space.

Week 6

• Discussion - 8th May

  • Topics: Regular Cell Complex and Cell Complex

    • We covered many examples of cell complex structure on spaces

Week 7

• Discussion - 15th May

  • Topics: Polygonal representation of surfaces, connected sums

    • Every compact surface can be expressed as a polygonal representation; a polygon with each side as a 1-cell

    • A polygonal representation of a compact surface tells one how to glue a single disc onto the 1-skeleten of the surface

Week 8 Quiz!! 📚

• Discussion - 22nd May

  • Topics: Cell Complex and connected sum review

    • Recall that a continuous map that is homeomorphic to its image is called an embedding

    • Regular cell complex structures on a space X require all k-cells to be embedded in in X, whereas cell complex allows identification of faces.

Week 9

• Discussion - 29th May

  • Topics:  

Week 10

• Discussion - 5th June

  • Topics: