Skip to main content

Exercises 1.4 Problem set 1, due 5 October 2021

1.

Prove or disprove: for every countable subspace \(X \subset \RR\) is discrete in the following sense: every subset \(S \subseteq X \) is both open and closed. [12 points]

2.

Let \(X \subseteq \RR^n\) be a subspace. For every pair of subsets \(S, T \in \PP(X)\text{,}\) write

\begin{equation*} d(S,T) = \inf\{ d(s,t) : (s \in S) \wedge (t \in T) \} . \end{equation*}

Suppose that \(S, T \in \PP(X)\) are subsets with the property that \(d(S,T) = 0\text{.}\) Prove or provide counterexamples for each of the following claims:

  • \(S \cap T \neq \varnothing\) [3 points];

  • \(S \cap \tau(T) \neq \varnothing\) [3 points];

  • \(S \cap \tau(T)) \cup (\tau(S) \cap T) \neq \varnothing\) [3 points];

  • \(\tau(S) \cap \tau(T) \neq \varnothing\) [3 points].

Define a subspace \(C \subset [0,1] \) as follows. For every natural number \(n \text{,}\) set

\begin{equation*} C_n \coloneq \bigcup_{k=0}^{3^{n-1}-1} \left(\left[\frac{3k}{3^n},\frac{3k+1}{3^n}\right]\cup\left[\frac{3k+2}{3^n},\frac{3k+3}{3^n}\right]\right) \comma \end{equation*}

and define

\begin{equation*} C \coloneq \bigcap_{n\geq 1} C_n \period \end{equation*}

The topological space \(C\) is called the Cantor space.

3.

Prove that \(C \) is closed in the interval \([0,1]\text{.}\) [12 points]

4.

What is the interior of \(C \) as a subspace of the interval \([0,1]\text{?}\) That is, what is the largest open subset \(U \subseteq [0, 1]\) such that \(U \subseteq C\text{?}\) [12 points]

Let \(X \subseteq \RR^m\) be a subspace. The formation of the closure is an operation

\begin{equation*} \tau \colon \PP(X) \to \PP(X) \end{equation*}

on the power set \(\PP(X)\) (i.e., a map from \(\PP(X)\) to itself). The formation of the complement is an operation

\begin{equation*} \kappa \colon \PP(X) \to \PP(X) \period \end{equation*}

Thus \(\kappa(S) = X \smallsetminus S\text{.}\) Please note that \(\tau\) is inclusion-preserving — that is, if \(S \subseteq T\text{,}\) then \(\tau(S) \subseteq \tau(T)\text{.}\) — and \(\kappa\) is inclusion-reversing — that is, if \(S \subseteq T\text{,}\) then \(\kappa(S) \supseteq \kappa(T)\text{.}\) Also of course \(S\subseteq \tau(S)\text{.}\) Finally, please observe that \(\tau\) is idempotent — that is, \(\tau^2 = \tau\) and that \(\kappa\) is involutive — that is, \(\kappa^2 = \operatorname{id}\text{.}\) We are interested in the operations \(\PP(X) \to \PP(X)\) that we can obtain by composing \(\tau\) and \(\kappa\) repeatedly. For example, the interior operator is

\begin{equation*} \iota \coloneq \kappa\tau\kappa \colon \PP(X) \to \PP(X). \end{equation*}

Note that \(\iota\) is inclusion-preserving, and \(\iota\) is idempotent. Many of the most important kinds of subsets of topological spaces are identified using \(\tau\) and \(\kappa\text{.}\) For example, a subset \(S\subseteq X\) is closed if and only if it is its own closure: \(S=\tau(S)=S\text{;}\) it is open if and only if it is its own interior: \(S=\iota(S)=\kappa\tau\kappa(S)\text{.}\)

5.

Write down all the subsets of \(\RR\) you can obtain by repeatedly applying the closure \(\tau\) and the interior \(\iota\) to the set

\begin{equation*} S \coloneq \{-30\} \cup \left]-20,0\right[ \cup \left]0,20\right[ \cup \left(\QQ \cap \left[25,30\right[\right). \end{equation*}

You do not need to prove that your computation is correct. [10 points]

6.

A subset \(S \subseteq X\) is said to be dense if \(\tau(S) = X\text{.}\) Find a countable dense subset of \(\RR\text{.}\) You do not need to prove that your computation is correct. [10 points]

7.

A subset \(S \subseteq X\) is said to be co-dense if it has empty interior, so that \(\iota(S) = \varnothing\text{.}\) Give an example of an uncountable co-dense subset \(S \subseteq \RR\text{.}\) You do not need to prove that your computation is correct. [10 points]

8.

A subset \(S \subseteq X\) is said to be nowhere dense if the interior of its closure is empty; that is, \(S\) is nowhere dense if \(\iota\tau(S)=\varnothing\text{,}\) or equivalently, \(\kappa\tau\kappa\tau(S) = \varnothing\text{.}\) Any nowhere dense subset of a topological space is co-dense, but give an example of a co-dense subset of \(\RR\) that is not nowhere dense. You do not need to prove that your computation is correct. [10 points]

9.

Show that if \(T\subseteq X\) is a closed co-dense subset, then any subset \(S \subseteq T\) is nowhere dense. [12 points]

10.

Optional. Let \(Z\subseteq X\text{.}\) Prove that \(Z\) is the closure of some open subset of \(X\) if and only if \(Z\) is the closure of its interior, so that \(Z=\tau\iota(Z)\text{,}\) or equivalently, \(Z = \tau\kappa\tau\kappa(Z)\text{.}\)

11.

Optional. Show that

\begin{equation*} \tau\kappa\tau = \tau\kappa\tau\kappa\tau\kappa\tau . \end{equation*}

Deduce that

\begin{equation*} \iota\tau = \iota\tau\iota\tau \text{ and } \tau\iota = \tau\iota\tau\iota . \end{equation*}
12.

Optional. Let \(S\subseteq X\text{.}\) What is the maximum number of sets one can form by repeatedly applying the closure and complement operators to \(S\text{?}\)