如何绘制$R^3$的乘积拓扑图？

Question

如何绘制 $R^3$ 的乘积拓扑图？如何绘制此图？

Answer 1

这是一个小的变化这个答案需要注意的一点是左对齐列，

/tikz/column 2/.style={nodes={align=left,text width={width("$\realR^3=\realR\times\realR\times\realR$")}}}

代码（带有两个不同的双笔画 R）

\documentclass{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{dsfont}
\newcommand{\realR}{\mathds{R}}
\usepackage{tikz-cd}
\begin{document}
\[\begin{tikzcd}[row sep=2em,column sep=2.5em,
    /tikz/column 2/.style={nodes={align=left,text width={width("$\realR^3=\realR\times\realR\times\realR$")}}}]
 & \realR & \\
 \mathcal{W} \arrow[ur,out=80,in=180,"g_1"] \arrow[dr,out=-80,in=180,"g_2"]
 \arrow[rr,out=-90,in=-110,"g_3",overlay] \arrow[r,dashed,"g"] 
 & \realR^3=\realR\times\realR\times\realR \arrow[r,"Pr_3"]
 & \realR\\
 & \realR & \\
\end{tikzcd}\]
Or with another double-stroke R\renewcommand{\realR}{\mathbb{R}}
\[\begin{tikzcd}[row sep=2em,column sep=2.5em,
    /tikz/column 2/.style={nodes={align=left,text width={width("$\realR^3=\realR\times\realR\times\realR$")}}}]
 & \realR & \\
 \mathcal{W} \arrow[ur,out=80,in=180,"g_1"] \arrow[dr,out=-80,in=180,"g_2"]
 \arrow[rr,out=-90,in=-110,"g_3",overlay] \arrow[r,dashed,"g"] 
 & \realR^3=\realR\times\realR\times\realR \arrow[r,"Pr_3"]
 & \realR\\
 & \realR & \\
\end{tikzcd}\]
\end{document}

附录：对 Zarko 答案的补充，其中箭头的移位是根据和真实 R 的宽度计算出来的inner sep。outer sep;-)

\documentclass{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{dsfont}
\newcommand{\realR}{\mathds{R}}
\usepackage{tikz-cd}
\begin{document}
\[\begin{tikzcd}[row sep=2em,column sep=2.5em,
    /tikz/column 2/.style={nodes={align=left,text width={width("$\realR^3=\realR\times\realR\times\realR$")}}}]
 & \realR & \\
 \mathcal{W} \arrow[ur,out=80,in=180,"g_1"] \arrow[dr,out=-80,in=180,"g_2"]
 \arrow[rr,out=-90,in=-110,"g_3",overlay] \arrow[r,dashed,"g"] 
 & \realR^3=\realR\times\realR\times\realR \arrow[r,"Pr_3"]
 % shift by half the width of the letter R + inner sep + outer sep (both equal 2pt)
 \arrow[u,start anchor={[xshift={width("$\realR$")/2+4pt}]north west},
    end anchor={[xshift={width("$\realR$")/2+4pt}]south west},"P_{p_1}"']
 \arrow[d,start anchor={[xshift={width("$\realR$")/2+4pt}]south west},
    end anchor={[xshift={width("$\realR$")/2+4pt}]north west},"P_{p_2}"]
 & \realR\\
 & \realR &\\
\end{tikzcd}\]
Or with another double-stroke R\renewcommand{\realR}{\mathbb{R}}
\[\begin{tikzcd}[row sep=2em,column sep=2.5em,
    /tikz/column 2/.style={nodes={align=left,text width={width("$\realR^3=\realR\times\realR\times\realR$")}}}]
 & \realR & \\
 \mathcal{W} \arrow[ur,out=80,in=180,"g_1"] \arrow[dr,out=-80,in=180,"g_2"]
 \arrow[rr,out=-90,in=-110,"g_3",overlay] \arrow[r,dashed,"g"] 
 & \realR^3=\realR\times\realR\times\realR \arrow[r,"Pr_3"]
 % shift by half the width of the letter R + inner sep + outer sep (both equal 2pt)
 \arrow[u,start anchor={[xshift={width("$\realR$")/2+4pt}]north west},
    end anchor={[xshift={width("$\realR$")/2+4pt}]south west},"P_{p_1}"']
 \arrow[d,start anchor={[xshift={width("$\realR$")/2+4pt}]south west},
    end anchor={[xshift={width("$\realR$")/2+4pt}]north west},"P_{p_2}"]
 & \realR\\
 & \realR & \\
\end{tikzcd}\]
\end{document}

Answer 2

作为@Schrödinger's cat 答案的补充。添加了标记为P_{r_1}和的垂直线P_{r_2}（在 OP 图像中显示，但未在他的答案中显示），并对图像代码做了一些小的更改：

\documentclass[margin=3mm,varwidth]{standalone}
\usepackage{tikz-cd}
\usepackage{amssymb}

\begin{document}
    \[
\begin{tikzcd}[row sep=huge,
/tikz/column 2/.style={nodes={text width = {width("$\mathbb{R}^3= \mathbb{R}\times\mathbb{R}\times\mathbb{R}$")}}},
                    ]
    & \mathbb{R} 
      \ar[d,start anchor={[xshift=1em]south west},  
              end anchor={[xshift=1em]north west}, "P_{r_1}",<-] 
                        &               \\
\mathcal{W} \ar[ur,bend left,"g_1"] 
            \ar[r,dashed,"g"]
            \ar[dr,bend right,"g_2"]
            \ar[rr,bend right=90,"g_3",looseness=1.5] 
    & \mathbb{R}^3=\mathbb{R}\times\mathbb{R}\times\mathbb{R} 
      \ar[r,"P_{r_3}"]  & \mathbb{R}    \\
    & \mathbb{R}
      \ar[u,start anchor={[xshift=1em]north west},
              end anchor={[xshift=1em]south west}, "P_{r_2}" ',<-]
                        &               \\
\end{tikzcd}
    \]    
\end{document}

Answer 3

我将在这里添加最差的答案，因为我不喜欢带有许多弯曲箭头或如此g_2接近的图表g_3。

\documentclass{article}

\usepackage{mathtools}
\usepackage{amssymb}
\usepackage{tikz-cd}

\begin{document}

\begin{tikzcd}
& \mathbb{R}^3\\
W\ar{ru}{g_1}\ar[dashed]{r}[description]{g}\ar{rd}[swap]{g_2}\ar[out=270,in=270]{rrr}
& \mathbb{R}^3\ar{u}[swap]{pr_1}\ar{d}{pr_2}
&[-11mm] =\mathbb{R}\times\mathbb{R}\times\mathbb{R}\ar{r}{pr_3}
&\mathbb{R}\\
& \mathbb{R}
\end{tikzcd}


\end{document}