Learning Materials
Features
Discover
Chapter 11: Problem 15
Give an example of disjoint closed sets \(F_{1}, F_{2}\) such that \(0=\inf \left\\{\left|x_{1}-x_{2}\right|: x_{i} \in F_{i}\right\\}\).
Over 30 million students worldwide already upgrade their learning with 91影视!
All the tools & learning materials you need for study success - in one app.
Get started for free
Using the dotation of the preceding exercise, let \(A, B\) be sets in \(\mathbb{R} .\) Show that \(A^{\circ} \subseteq A,\left(A^{\circ}\right)^{\circ}=\) \(A^{\circ}\), and that \((A \cap B)^{\circ}=A^{\circ} \cap B^{\circ} .\) Show also that \(A^{\circ} \cup B^{c} \subseteq(A \cup B)^{\circ}\), and give an example to show that the inclusion may be proper.
Show that if \(G\) is an open set and \(F\) is a closed set, then \(G \backslash F\) is an open set and \(F \backslash G\) is a closed set.
Let \(\left(K_{n}: n \in \mathbb{N}\right)\) be a sequence of nonempty compact sets in \(\mathbb{R}\) such that \(K_{1} \supseteq K_{2} \supseteq \cdots \supseteq\) \(K_{n} \supseteq \cdots\) Prove that there exists at least one point \(x \in \mathbb{R}\) such that \(x \in K_{n}\) for all \(n \in \mathbb{N} ;\) that is, the intersection \(\bigcap_{n=1}^{\infty} K_{n}\) is not empty.
Show that if \(f: \mathbb{R} \rightarrow \mathbb{R}\) is continuous. then the set \(\\{x \in \mathbb{R}: f(x) \leq \alpha\\}\) is closed in \(\mathbb{R}\) for each \(\alpha \in \mathbb{R}\)
Give an example of a function \(f: \mathbb{R} \rightarrow \mathbb{R}\) such that the set \(\\{x \in \mathbb{R}: f(x)=1\\}\) is neither open nor closed in \(\mathbb{R}\).
What do you think about this solution?
We value your feedback to improve our textbook solutions.
Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine