Practice test 1

(Also available as postscript and acrobat file off my web page).

The test on Thursday will be open book - just the book, nothing else is permitted (and no notes in your book!) Note that where $ \mathbb{R}^k$ is mentioned below the standard metric is assumed.

  1. Let $ C\subset\mathbb{R}^n$ be closed. Show that there is a point $ p\in C$ such that $ \vert p\vert=\inf\{\vert x\vert;x\in X\}.$
  2. Give a counterexample to each of the following statements:
    1. Subsets of $ \mathbb{R}$ are either open or closed
    2. A closed and bounded subset of a metric space is compact.
    3. In any metric space the complement of a connected set is connected.
    4. Given a sequence in a metric space, if every subsequence of that sequence itself has a convergent subsequence then the original sequence converges.
  3. Suppose $ A$ and $ B$ are connected subsets of a metric space $ X$ and that $ A\cap B\not=\emptyset,$ show that $ A\cup B$ is connected.
  4. Let $ K_i,$ $ i=1,\dots,N,$ be a finite number of compact sets in a metric space $ X.$ Show that $ \bigcup_{i=1}^NK_i$ is compact.
  5. Let $ G_i\subset X,$ $ i\in\mathbb{N}$ be a countable collection of open subsets of a complete metric space, $ X.$ Suppose that for each $ N\in\mathbb{N},$ $ \bigcap_{i=1}^NG_i\not=X$ and that for each $ n,$ $ x,y\in
X\setminus G_n\Longrightarrow d(x,y)<1/n.$ Show that $ \bigcup_{i\in\mathbb{N}}G_i\not=X.$
  6. Let $ x_n$ be a bounded sequence in $ \mathbb{R}.$ Show that there exists $ x\in\mathbb{R}$ and a subsequence $ x_{n(k)}$ such that $ \sum\limits_{k=1}^\infty x_{n(k)}-x$ converges absolutely.

Richard B. Melrose 2004-05-18