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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