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<title>MathJax example</title>
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src="https://cdn.jsdelivr.net/npm/mathjax@4/tex-mml-chtml.js">
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Latex Code Here
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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>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width">
<title>MathJax example</title>
<script id="MathJax-script" async
src="https://cdn.jsdelivr.net/npm/mathjax@4/tex-mml-chtml.js">
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\(\textbf{B.5} \) 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.
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\[\sin(x)= \sum_{n \mathop = 0}^\infty ( -1)^n \frac {x^{2 n + 1} } {\left( {2 n + 1}\right)!}\]
<br><br><br>
\(\begin{bmatrix}
1&0&0&x\\
0&1&0&y\\
0&0&1&z
\end{bmatrix}
\cdot
\begin{bmatrix}
1&0&0\\
0&1&0\\
0&0&1\\
x&y&z
\end{bmatrix}\)
<br><br><br>
\[\left\{\begin{array}{lc}
1&0&0&x\\
0&1&0&y\\
0&0&1&z
\end{array}\right\}\]
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<script type="text/javascript" async src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-MML-AM_CHTML"></script> <center> \(Latex\) and Text Here </center>
<script type="text/javascript" async src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-MML-AM_CHTML"></script> <center>\[
f_r:\{r\}\longrightarrow \{0,1\}\\
r\longmapsto \left\{\begin{array}{lc}
1,&r\in\mathbb{Q} \\
0,&r\notin \mathbb{Q}
\end{array}\right.
\]</center>
<script type="text/javascript" async src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-MML-AM_CHTML"></script> <left>
\(\textbf{B.5} \) 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.
<br><br><br>
\[\sin(x)= \sum_{n \mathop = 0}^\infty ( -1)^n \frac {x^{2 n + 1} } {\left( {2 n + 1}\right)!}\]
<br><br><br>
\(\begin{bmatrix}
1&0&0&x\\
0&1&0&y\\
0&0&1&z
\end{bmatrix}
\cdot
\begin{bmatrix}
1&0&0\\
0&1&0\\
0&0&1\\
x&y&z
\end{bmatrix}\)
<br><br><br>
\[\left\{\begin{array}{lc}
1&0&0&x\\
0&1&0&y\\
0&0&1&z
\end{array}\right\}\]
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