General Topology

• State the definition of compact and all "smaller" versions of compact: locally compact, sequentially compact, etc., along with their definitions. (Sarason)
• Is the Arzela-Ascoli Theorem reversible? Why or why not? (Sarason)
• Define limit point, accumulation point and cluster point. (Sarason)
• Why or why not is the definition of compact equivalent to the definition of complete? Give examples of spaces which are one but not the other, or prove one implies the other if given a specific type of space. (Sarason)
• In what kind of space is a unique cluster point a limit point? (Sarason)
• What do you think is a reasonable definition of a boudned metric space? Prove that every compact metric space is bounded under that definition. (Arveson)
• Talk about the Baire Category Theorem; give an application. (Sarason)
• Describe the Uniform Boundedness Principle. (Sarason)
• Talk about the Weierstrass Approximation Theorem. Is the set of continuous functions from a compact subset of to the reals a separable set? (Sarason)
• Define various forms of compactness. What implications hold among these? Give an example of a compact space which is not sequentially compact and vice-versa. (Sarason)
• State equivalent formulations of compactness in metric spaces. Prove any one of the nontrivial implications among these characterizations. Characterize closed subsets of . (Sarason)
• Define connectedness and path connectedness. Give an example of a connected space which is not path connected. (Sarason)
  • Give a compact example. (Bergman)
  • Show that a connected open subset of is path connected. (Sarason)

This page was originally derived from this TeX file.

These are the questions I was asked on topology as part of my qual. The syllabus for the topology part can be found on the Logic Page.

• Give an example of a T₀ space which isn't T₁. What are the other separation axioms? Conjecture and prove classification theorems for finite T₂, T₁ and T₀ spaces.

• State and prove Tychonoff's theorem. What's it equivalent to over ZF? How does your answer change if you take the definition of compact to include T₂?

--Abooth 9/2.