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Next: Problem 3 Up: 18.100B, Fall 2002 In-class Previous: Problem 1

Problem 2

Consider the metric space which is the subset $ E=\{0\}\cup\bigcup_{n\in\mathbb{N}}\{1/n\}$ of the real numbers with the metric induced by the usual metric on $ \mathbb{R}.$
  1. What is the set $ E'$ of limit points of $ E,$ in $ \mathbb{R}$?
  2. Describe all the closed subsets of $ E.$
  3. Describe all the compact subsets of $ E.$
  4. Describe all the connected subsets of $ E.$
In each case justify your answer.

Proof. [Solution]
  1. 0 is the only limit point. It is a limit point since $ 1/n\to0$ as $ n\to\infty$ and all other points are isolated, so cannot be limit points.
  2. A subset $ C\subset E$ is closed if it is finite or contains 0 (or both). To see this, recall that $ C$ is closed in $ E$ if and only if it contains all its limit points. A limit point of $ C$ must be a limit point of $ E$ so the only possibilities are that $ C$ has no limit points or it contains $ 0.$ In the latter case it is closed and the former means that 0 is either not in $ C$ and is not a limit point of $ C,$ which therefore must be finite.
  3. Since $ E$ contains its only limit point in $ \mathbb{R}$ it is closed and bounded in $ \mathbb{R},$ so it is compact by the Heine-Borel Theorem. Thus the compact subsets of $ E$ are just the closed sets as described above, since as shown in class a subset of a compact space is compact if and only if it is closed.
  4. Let $ G\subset E$ be a connected set. If $ p=1/n\in G$ is one of the isolated points of $ E$ then $ G=A\cup B,$ $ A=\{p\},$ $ B=G\setminus\{p\}$ is a decomposition with $ \bar A=A$ and $ p\notin\bar B$ so $ \bar A\cap
B=\emptyset$ and $ \bar A\cap B=\emptyset.$ Thus $ B=\emptyset$ by the definition of connectedness and $ G=\{p\}.$ Thus the only possibility is that $ G$ consists of any one point of $ E$ and these sets are trivially connected since in any decompostion $ \{q\}=A\cup B$ with $ A\cup
B=\emptyset$ one of $ A$ or $ B$ must be empty.
$ \qedsymbol$

next up previous
Next: Problem 3 Up: 18.100B, Fall 2002 In-class Previous: Problem 1
Richard B. Melrose 2002-10-19