数学 – 位相空間 - 位相空間(3点より成る集合、すべての位相)

集合・位相入門

学習環境

• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Internet Explorer, Google Chrome...)用JavaScript Library: MathJax

集合・位相入門(松坂 和夫(著)、岩波書店)の第4章(位相空間)、2(位相空間)、問題1.を解いてみる。

問1.

$\begin{array}{l}\mathfrak{P}\left(S\right)=\left\{\varphi ,\left\{p\right\},\left\{q\right\},\left\{r\right\},\left\{p,q\right\},\left\{q,r\right\},\left\{r,p\right\},S\right\}\\ {\mathfrak{O}}_1=\left\{\varphi ,S\right\}\\ {\mathfrak{O}}_2=\left\{\varphi ,\left\{p\right\},S\right\}\\ {\mathfrak{O}}_3=\left\{\varphi ,\left\{q\right\},S\right\}\\ {\mathfrak{O}}_4=\left\{\varphi ,\left\{r\right\},S\right\}\\ {\mathfrak{O}}_5=\left\{\varphi ,\left\{p,q\right\},S\right\}\\ {\mathfrak{O}}_6=\left\{\varphi ,\left\{q,r\right\},S\right\}\\ {\mathfrak{O}}_7=\left\{\varphi ,\left\{r,p\right\},S\right\}\\ {\mathfrak{O}}_8=\left\{\varphi ,\left\{p\right\},\left\{p,q\right\},S\right\}\\ {\mathfrak{O}}_9=\left\{\varphi ,\left\{p\right\},\left\{p,r\right\},S\right\}\\ {\mathfrak{O}}_{10}=\left\{\varphi ,\left\{p\right\},\left\{q,r\right\},S\right\}\\ {\mathfrak{O}}_{11}=\left\{\varphi ,\left\{q\right\},\left\{q,r\right\},S\right\}\\ {\mathfrak{O}}_{12}=\left\{\varphi ,\left\{q\right\},\left\{q,p\right\},S\right\}\\ {\mathfrak{O}}_{13}=\left\{\varphi ,\left\{q\right\},\left\{r,p\right\},S\right\}\\ {\mathfrak{O}}_{14}=\left\{\varphi ,\left\{r\right\},\left\{r,p\right\},S\right\}\\ {\mathfrak{O}}_{15}=\left\{\varphi ,\left\{r\right\},\left\{r,q\right\},S\right\}\\ {\mathfrak{O}}_{16}=\left\{\varphi ,\left\{r\right\},\left\{p,q\right\},S\right\}\\ {\mathfrak{O}}_{17}=\left\{\varphi ,\left\{p\right\},\left\{q\right\},\left\{p,q\right\},S\right\}\\ {\mathfrak{O}}_{18}=\left\{\varphi ,\left\{q\right\},\left\{r\right\},\left\{q,r\right\},S\right\}\\ {\mathfrak{O}}_{19}=\left\{\varphi ,\left\{r\right\},\left\{p\right\},\left\{r,p\right\},S\right\}\\ {\mathfrak{O}}_{20}=\left\{\varphi ,\left\{p\right\},\left\{p,q\right\},\left\{p,r\right\},S\right\}\\ {\mathfrak{O}}_{21}=\left\{\varphi ,\left\{q\right\},\left\{p,q\right\},\left\{q,r\right\},S\right\}\\ {\mathfrak{O}}_{22}=\left\{\varphi ,\left\{r\right\},\left\{p,r\right\},\left\{q,r\right\},S\right\}\\ {\mathfrak{O}}_{23}=\left\{\varphi ,\left\{p\right\},\left\{q\right\},\left\{p,q\right\},\left\{p,r\right\},S\right\}\\ {\mathfrak{O}}_{24}=\left\{\varphi ,\left\{p\right\},\left\{q\right\},\left\{p,q\right\},\left\{q,r\right\},S\right\}\\ {\mathfrak{O}}_{25}=\left\{\varphi ,\left\{p\right\},\left\{r\right\},\left\{p,r\right\},\left\{p,q\right\},S\right\}\\ {\mathfrak{O}}_{26}=\left\{\varphi ,\left\{p\right\},\left\{r\right\},\left\{p,r\right\},\left\{q,r\right\},S\right\}\\ {\mathfrak{O}}_{27}=\left\{\varphi ,\left\{q\right\},\left\{r\right\},\left\{q,r\right\},\left\{q,p\right\},S\right\}\\ {\mathfrak{O}}_{28}=\left\{\varphi ,\left\{q\right\},\left\{r\right\},\left\{q,r\right\},\left\{p,r\right\},S\right\}\\ {\mathfrak{O}}_{29}=\mathfrak{P}\left(S\right)\end{array}$