Exercise 3.2.9

Question

[De Morgan's Laws]
  A proof for De Morgan's Laws in the case of two sets is outlined in Exercise 1.2.5. The general argument is similar.
  \begin{enumerate}[label=(\alph*)] 
  \item Given a collection of sets $\left\{E_{\lambda}: \lambda \in \Lambdaight\}$, show that
    $$
    \left(\bigcup_{\lambda \in \Lambda} E_{\lambda}ight)^{c}=\bigcap_{\lambda \in \Lambda} E_{\lambda}^{c} \quad 	ext { and } \quad\left(\bigcap_{\lambda \in \Lambda} E_{\lambda}ight)^{c}=\bigcup_{\lambda \in \Lambda} E_{\lambda}^{c}
    $$
  \item Now, provide the details for the proof of Theorem 3.2.14.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item If $x \in \left(\bigcup_{\lambda \in \Lambda} E_\lambdaight)^c$ then $x 
otin \bigcup_{\lambda \in \Lambda} E_\lambda$ meaning $x 
otin E_\lambda$ for all $\lambda \in \Lambda$ implying $x \in E_\lambda^c$ for all $\lambda\in\Lambda$ and so finally $x \in \bigcap_{\lambda\in\Lambda} E_\lambda^c$. This shows
    $$
    \left(\bigcup_{\lambda\in\Lambda}E_\lambdaight)^c \subseteq \bigcap_{\lambda\in\Lambda} E_\lambda^c
    $$
    To show the reverse inclusion suppose $x \in \bigcap_{\lambda\in\Lambda} E_\lambda^c$ then $x 
otin E_\lambda$ for all $\lambda$ meaning $x 
otin \bigcup_{\lambda\in\Lambda} E_\lambda$ and so the reverse inclusion
    $$
    \bigcap_{\lambda\in\Lambda} E_\lambda^c \subseteq \left(\bigcup_{\lambda\in\Lambda}E_\lambdaight)^c
    $$
    Is true, completing the proof.
  \item
    Let $F = F_1 \cup F_2$, if $x_n \in F$ and $x = \lim x_n$. Let $(x_{n_k})$ be a subsequence of $(x_n)$ fully contained in $F_1$ or $F_2$. the subsequence $(x_{n_k})$ must also converge to $x$, meaning $x$ is in $F_1$ or $F_2$, the rest is by induction.

Now let $F = \bigcap_{\lambda\in\Lambda} F_\lambda$
    $$
    F^c = \bigcup_{\lambda\in\Lambda} F_\lambda^c
    $$
    Each $F_\lambda^c$ is open by Theorem 3.2.13, thus Theorem 3.2.3 (ii) implies $F^c$ is open, and so $(F^c)^c = F$ is closed.
   \end{enumerate}

Exercise 3.2.9

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Exercise 3.2.9

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