\( \def\naturals{\mathbb{N}} \def\integers{\mathbb{Z}} \def\rationals{\mathbb{Q}} \def\reals{\mathbb{R}} \def\complex{\mathbb{C}} \)
Topics: Metric Spaces
Book Chapters: Rudin, Chapter
Due date: Before class, 5 Oct 2015
Mode of submissions: Please submit at blackboard.
Format: Please keep it in pdf. format with your name and andrew id included in it. The filename must be LastName_FirstName-yourAndrewID-HW2.
Expected time required for this homework: Six hours. If you're quick, you can do it in under an hour.

I. DeMorgan's Laws

  1. Let $\{E_\alpha\}$ be a collection of sets in any metric space. The subscript $\alpha$ is used to indicate that the collection may not be countable. Prove the following. The superscript $c$ represents a complement relative to the metric space.
    1. \[ \left( \bigcup_\alpha E_\alpha\right)^c = \bigcap_\alpha E_\alpha^c \] Hint: You may find it easier to prove that if any element $x$ belongs to the LHS implies that it belongs to the RHS and vice versa.
    2. \[ \left(\bigcap_\alpha E_\alpha\right)^c = \bigcup_\alpha E_\alpha^c \]
  2. Prove the following.
    1. The arbitrary union of open sets is open. Hint: Recall that an open set, by definition, is one in which every element is an interior point.
    2. The arbitrary intersection of closed sets is closes. Hint: De Morgan's laws.
    3. Finite intersection of open sets is open.
    4. Finite union of closed sets is closed.

II. Open and Closed Sets

  1. Show that
    1. The (relative) complement of an open set is closed.
    2. The (relative) complement of closed set is open.
  2. Consider any set $C$. Let $C^\prime$ be the set of limit points of $C$. The closure of $C$ is defined as $\bar{C} = C \cup C^\prime$. Show that $\bar{C}$ is also closed.

III. Compact Sets

Show the following. In general, for proofs relating to compactness, we draw upon the fact than any cover has a finite subcover, and we now only have deal with a finite number of elements. Recall also that finite collections of numbers provably have a supremum and an infimum within the collection.

  1. Finite sets are compact.
  2. Compact sets are bounded.
  3. Compact sets are closed.
  4. For any set $C$ (in a metric space) \[ C \textrm{ compact} \Longleftrightarrow C \textrm{ closed and bounded} \] Note that you have to prove this both ways, as this is a bidirectional relation.
  5. If $C$ is closed and $K$ is compact, $C \cap K$ is compact.