# Heartbeat

I have not written in a while, but this blog is still alive, so I will quickly prove two of my favorite (among many) results of topology. These stem straight from my preparation for the coming preliminary examinations.

School all of a sudden snatched most of my time.

Let $X$ be a topological space.

**Claim**: $X$ is Hausdorff if and only if $\Delta:=\left\{\left(x,x\right):x\in X\right\}$ is closed.

**Proof**: Suppose that $X$ is Hausdorff, and let $\left(x_1,x_2\right)\in X\times X\setminus\Delta$. Then $x_1\neq x_2$, which implies that there exist disjoint neighborhoods $U$ and $V$ of $x_1$ and $x_2$, respectively, which implies that $U\times V\subseteq X\times X\setminus\Delta$, which implies that $X\times X\setminus\Delta$ is open, which implies that $\Delta$ is closed.

Suppose that $\Delta$ is closed, and let $x_1,x_2\in X$ such that $x_1\neq x_2$. Then $\left(x_1,x_2\right)\in X\times X\setminus\Delta$, which is open, which implies that there exists a neighborhood $U\times V$ of $\left(x_1,x_2\right)$ contained in $X\times X\setminus\Delta$, which implies that there exist disjoint neighborhoods $U$ and $V$ of $x_1$ and $x_2$, respectively, which implies that $X$ is Hausdorff. $\blacksquare$

Let $X$ and $Y$ be topological spaces, let $Y$ be Hausdorff, let
$A\subseteq X$, let $f,g:\overline A\to Y$ be continuous, and let
$f\left(x\right)=g\left(x\right)$ for every $x\in A$.

**Claim**: $f\left(x\right)=g\left(x\right)$ for every $x\in\overline A$.

**Proof**: Define $h:\overline A\to Y\times Y$ by $x\mapsto\left(f\left(x\right),g\left(x\right)\right)$. Then $h$ is continuous and $h\left(A\right)\subseteq\Delta$, which implies that $h(\overline A)\subseteq\overline{h\left(A\right)}\subseteq\overline\Delta$. Since $Y$ is Hausdorff, $\Delta$ is closed, which implies that $h(\overline A)\subseteq\Delta$, which implies that $f\left(x\right)=g\left(x\right)$ for every $x\in\overline A$. $\blacksquare$

I like these results because the first characterizes Hausdorff spaces in simple terms, and the second shows that no shenanigans happen at the continuous images in Hausdorff spaces of limit points.

**Remark**: It is implicit in the first claim that the topology on $X\times X$ is the product one, which is generated by the set of products of open subsets of $X$. Let $\left(x_1,x_2\right)\in X\times X$, and let $N$ be a neighborhood of $\left(x_1,x_2\right)$. Then

$$N=\bigcup_{\alpha\in I}U_\alpha\times V_\alpha,$$

where $U_\alpha$ and $V_\alpha$ are open, and $I$ is some index set. Therefore, there exists a $\beta\in I$ such that $\left(x_1,x_2\right)\in U_\beta\times V_\beta$.

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