If $A$ is clopen in $X$ then $\overline{A}$ is clopen in $\beta X$.

Question

Let $X$ be a Tychonoff space. If $A$ is a clopen subset of $X$, then $\overline{A}$ is clopen in $\beta X$ (of course, $\overline{A}$ means the closure of $A$ in $\beta X$).

This is actually the Corollary 3.6.5 from Engelking, but he gives no proof as if it were immediately. Here is a pic: https://i.stack.imgur.com/mqV42.png

Most likely it is related to the corollary that states every continuous function $f:X\to [0,1]$ has a continuous extension.

Can anyone give me a hint to show $\overline{A}$ is open?

Answer 1

HINT: Look at the continuous function $f:X\to\{0,1\}$ that takes every point of $A$ to $0$ and every point of $X\setminus A$ to $1$.