Problem Set 5

  1. Easy
    Show directly from the definition that the following sets are disconnected:
    1. The hyperbola \(\{(x,y)\in\mathbb{R}^{2}: x^2-y^2 = 1\}\)
    2. Any finite set of points in \(\mathbb{R}^{n}\) with more than two elements
    3. \(\{(x,y,z)\in\mathbb{R}^{3} : xyz > 0\}\)

  2. Easy
    Let \(\left\{S_{n}\right\}_{n=1}^{\infty}\) be a collection of path connected sets with the property that \(S_{n}\cap S_{n+1} \neq \emptyset\) for every \(n\). Prove that \(\bigcup_{n=1}^{\infty} S_{n}\) is connected.

  3. Easy
    If \(A\) and \(B\) are connected, must \(A\cup B\) be connected? Must \(A\cap B\) be connected? Provide a proof or a counterexample in both cases. If \(A\) and \(B\) are convex, is \(A\cap B\) convex?

  4. Medium
    Prove that if \(A\subseteq \mathbb{R}^{m}\) and \(B\subseteq \mathbb{R}^{n}\) are compact sets, then \(A\times B = \{(x,y) \in\mathbb{R}^{n+m}:x\in A, y\in B\}\) is compact.

  5. Medium
    Suppose that \(f: S\to\mathbb{R}\) is continuous, and let \(L = \inf\left\{ f(x) : x \in S\right\}\). Can we always guarantee that there exists \(y \in S\) such that \(f(y) = L\)? Prove the statement, or else provide a counterexample. If the statement is false, how could we have modified the problem to make it true?

  6. Medium
    The distance between two sets \(U,V\subseteq\mathbb{R}^{n}\) is defined by: \[ d(U,V) = \inf\left\{\vert x-y\vert:x\in U, y\in V\right\} \]
    1. Show that \(d(U,V) = 0\) if either \(\exists\) \(x \in \overline{U}\cap V\) or \(\exists\) \(x \in \overline{V}\cap U\)
    2. Show that if \(U\) is compact, \(V\) is closed, and \(U\cap V = \emptyset\), then \(d(U,V) > 0\).
    3. Show that the compactness of \(U\) in the previous part was necesssary by giving an example of two closed sets \(U\) and \(V\) in \(\mathbb{R}^{2}\) which share no point in common, but satisfy \(d(U,V) = 0\).

  7. Medium
    Prove that if \(K_1\supset K_2 \supset K_3 \supset K_4 \supset \ldots\) is a chain of proper containments and each \(K_{i}\subseteq \mathbb{R}^{n}\) is compact, then \(\bigcap_{i=1}^{\infty} K_{i} \neq \emptyset\)

  8. Medium
    Show that \(\{(x,y,z)\in\mathbb{R}^{3}:x^2+y^2+z^2 = 1\}\) is connected.

  9. Medium
    Let \(A_i\) be a compact subset of \(\mathbb R^n\) for each \(i \in \mathbb N\). Prove or provide a counter example for each of the following statements.
    1. Finite unions \( \bigcup_{k=1}^m A_i \) are compact.
    2. Infinite unions \( \bigcup_{k=1}^\infty A_i \) are compact.
    3. Finite intersections \( \bigcap_{k=1}^m A_i \) are compact.
    4. Infinite intersections \( \bigcap_{k=1}^\infty A_i \) are compact.

  10. Hard
    Let \(U\subseteq \mathbb{R}^{n}\) be an open set. We say that \(f: U \to \mathbb{R}\) is proper if and only if \(f\) is continuous, and for every compact set \(K\subseteq \mathbb{R}\), \(f^{-1}(K)\subseteq U\) is compact. Suppose that we have a finite set \(A \subseteq \mathbb{R}^{n}\) and a continuous function \(f: \mathbb{R}^{n}\backslash A \to \mathbb{R}\) such that for every \(a \in A\), \[ \lim_{x\to a} \vert f(x)\vert = \lim_{x\to\infty} \vert f(x)\vert = \infty \] Show that \(f\) is proper.

  11. Hard
    Let \(S\subseteq \mathbb{R}^{n}\). \(S\) is disconnected if and only if there exists a continuous function \(f:S\to\mathbb{R}\) such that \(f(S) = \left\{0,1\right\}\).

  12. Hard
    Let \(K \subseteq \mathbb R^n\) be a compact set, and \(f: K \to \mathbb R^m \) be a continuous function. Show that for any fixed \(\epsilon >0\) there exists a \(\delta >0\) such that for all \(x,y\in K\) we have \( \|x-y\|<\delta \Rightarrow \|f(x)-f(y)\|< \epsilon\). [Note that in the case of a continuous function, the choice of \(\delta\) is allowed to depend upon \(x,y\). Here we are trying to find a single \(\delta\) which works for all \(x,y\)].

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