M1 Midterm 1 Review

What you need to know for the midterm.

1. Topological spaces

(Munkres ch. 2, §12–21)

• Topological space: , . Example: metric spaces.
• Axioms: , , arbitrary unions, and finite intersections of open subsets are open. is closed is open.
• Basis for a topology:
  • open sets unions of elements of . is open for all there exists such that .
  • axioms for a basis: ; there exists such that .
  • Example: the open balls in a metric space are a basis for the metric topology.
• If , say is finer and is coarser.
• is continuous if for all open , is open. (For metric spaces, this is for all and all there exists such that .)
• Closure: all closed subsets . Interior: all open subsets . , and every neighborhood of intersects .
• Limit points of subsets ( is a limit point of for every neighborhood , ) are not the same as limits of sequences ( for every neighborhood , all but finitely many ). (But they are related in metric spaces: is closed whenever with , then .)
• Subspace topology on : .
• Product topology: basis .
  • For products of metric spaces, the uniform topology (, up to truncation) is finer.
  • is continuous in the product topology iff each is continuous.
  • For : in the product topology pointwise convergence, for all ; in the uniform topology uniform convergence, .
  • uniformly, continuous continuous.
• is Hausdorff if for all there exist open , such that . ( uniqueness of limits of sequences.) Metric spaces are Hausdorff.

2. Connectedness and compactness

(Munkres ch. 3, §23–24, 26–29)

• is connected if with open and disjoint implies one is and the other is .
• continuous, connected connected. ( the intermediate value theorem, since the connected subsets of are the intervals.)
• Path-connected: any two points of can be joined by a (continuous) path . Path-connected connected ( in general).
• is compact if for every open cover there exists a finite subcover .
• continuous, compact compact ( the extreme value theorem).
• compact, closed compact. Hausdorff, compact closed. In , compact closed and bounded.
• (Finite) products of connected spaces are connected; (finite) products of compact spaces are compact.
• If is a compact metric space, then:
  • every open cover has a Lebesgue number : there exists such that ;
  • every continuous function is uniformly continuous.
• For metric spaces: compact (open covers have finite subcovers) limit point compact (every infinite subset has a limit point) sequentially compact (every sequence has a convergent subsequence).
• Compact metric spaces are complete (Cauchy sequences converge). and with the uniform topology are complete.
• A one-point compactification of is a compact space such that (homeomorphic).
  • Build: , opens of complements of compact subsets of .
  • If is locally compact (for all there exist a neighborhood and a compact ) and Hausdorff, then is Hausdorff and unique up to homeomorphism.