Thomas Carlson - MathJAX Cheat Sheet

How to embed LATEX code on Google Sites with MathJAX

Way #1

<html>

<head>

<title>MathJax example</title>

src="https://cdn.jsdelivr.net/npm/mathjax@4/tex-mml-chtml.js">

</script>

</head>

<body>

<p>

Latex Code Here

</p>

</body>

You must use  or \[ \], not $ $ or $$ $$.

No environments or packages, only standard Latex commands.

\\ works in math environment, <br> is html command for new line.

<html>

<head>

<title>MathJax example</title>

src="https://cdn.jsdelivr.net/npm/mathjax@4/tex-mml-chtml.js">

</script>

</head>

<body>

<p>

$\textbf{B.5} $ Let $M$ be a metric space. <br> (a.) Both $M$ and $\emptyset$ are open subsets of $M$. <br> (b.) Any intersection of finitely many open sets in $M$ is open in $M$. <br> (c.) Any union of arbitrarily many open sets in $M$ is open in $M$. <br> <br>

$\textbf{Proof} $<br><br>$(a.)$ The empty set has no points, so it is vacuously open. Given any point $x\in M$, $B_1(x)$ is an open ball in $M$ which contains $x$, so $M$ is an open set. <br><br>

$(b.)$ Given $\{U_i\}_{i=1}^n$ a finite collection of open subsets of $M$, let $m\in \bigcap_{i=1}^{n} U_i.$ Then, as each $U_i$ is open, we have a collection $\{B_{r_i}(m)\}$ of open balls in $M$, where $B_{r_i}(m)\subset U_i.$ Call the smallest radius $r_0$, which exists as there are finitely many $r_i$. Then as a ball is contained in any ball centered at the same point with a larger radius, $B_{r_0}(m)$ is therefore contained in every $U_i$. Therefore $B_{r_0}(m)$ is an open ball around $m$ which is contained in the intersection $\bigcap_{i=1}^{n} U_i$, so finite intersections of open sets are indeed open. <br><br>

$(c.)$ Given a collection $\{U_i\}$ of open sets, let $m\in \bigcup_{i} U_i$ . Then, in particular, there exists $U_k$ such that $m\in U_k$ , which is open. Therefore, there is a ball around $m$ in $U_k$ , which is contained in the union as well, so the union is indeed open.

\[\sin(x)= \sum_{n \mathop = 0}^\infty ( -1)^n \frac {x^{2 n + 1} } {\left( {2 n + 1}\right)!}\]

\(\begin{bmatrix}

1&0&0&x\\

0&1&0&y\\

0&0&1&z

\end{bmatrix}

\cdot

\begin{bmatrix}

1&0&0\\

0&1&0\\

0&0&1\\

x&y&z

\end{bmatrix}\)

\[\left\{\begin{array}{lc}

1&0&0&x\\

0&1&0&y\\

0&0&1&z

\end{array}\right\}\]

</p>

</body>

</html>

Way #2

f_r:\{r\}\longrightarrow \{0,1\}\\

r\longmapsto \left\{\begin{array}{lc}

1,&r\in\mathbb{Q} \\

0,&r\notin \mathbb{Q}

\end{array}\right.

\]</center>

$\textbf{B.5} $ Let $M$ be a metric space. <br> (a.) Both $M$ and $\emptyset$ are open subsets of $M$. <br> (b.) Any intersection of finitely many open sets in $M$ is open in $M$. <br> (c.) Any union of arbitrarily many open sets in $M$ is open in $M$. <br> <br>

$\textbf{Proof} $<br><br>$(a.)$ The empty set has no points, so it is vacuously open. Given any point $x\in M$, $B_1(x)$ is an open ball in $M$ which contains $x$, so $M$ is an open set. <br><br>

$(b.)$ Given $\{U_i\}_{i=1}^n$ a finite collection of open subsets of $M$, let $m\in \bigcap_{i=1}^{n} U_i.$ Then, as each $U_i$ is open, we have a collection $\{B_{r_i}(m)\}$ of open balls in $M$, where $B_{r_i}(m)\subset U_i.$ Call the smallest radius $r_0$, which exists as there are finitely many $r_i$. Then as a ball is contained in any ball centered at the same point with a larger radius, $B_{r_0}(m)$ is therefore contained in every $U_i$. Therefore $B_{r_0}(m)$ is an open ball around $m$ which is contained in the intersection $\bigcap_{i=1}^{n} U_i$, so finite intersections of open sets are indeed open. <br><br>

$(c.)$ Given a collection $\{U_i\}$ of open sets, let $m\in \bigcup_{i} U_i$ . Then, in particular, there exists $U_k$ such that $m\in U_k$ , which is open. Therefore, there is a ball around $m$ in $U_k$ , which is contained in the union as well, so the union is indeed open.

\[\sin(x)= \sum_{n \mathop = 0}^\infty ( -1)^n \frac {x^{2 n + 1} } {\left( {2 n + 1}\right)!}\]

\(\begin{bmatrix}

1&0&0&x\\

0&1&0&y\\

0&0&1&z

\end{bmatrix}

\cdot

\begin{bmatrix}

1&0&0\\

0&1&0\\

0&0&1\\

x&y&z

\end{bmatrix}\)

\[\left\{\begin{array}{lc}

1&0&0&x\\

0&1&0&y\\

0&0&1&z

\end{array}\right\}\]

</left>