A Urysohn Function on an Uncountable Fort Space

Question

I'm struggling with a problem about determining if an Uncountable Fort Space is Tychonoff or not, at this moment I have proven that my space is regular Hausdorff but I can't find a Urysohn Function for the space in question.

Any hint or tip is welcome :)

Answer 1

It’s a compact Hausdorff space, so it is not just Tikhonov but $T_4$ (normal plus $T_1)$.

However, we can also show the result directly. Let $X$ be the space, and let $p$ be the non-isolated point. The closed sets in $X$ are the sets containing $p$ and the finite subsets of $X\setminus\{p\}$.

Suppose first that $p\in F\subseteq X$ and $x_0\in X\setminus F$. Then $\{x_0\}$ is a clopen set, so the function

$$f:X\to[0,1]:x\mapsto\begin{cases} 0,&\text{if }x=x_0\\ 1,&\text{otherwise} \end{cases}$$

is continuous, $f(x_0)=0$, and $f[F]=\{1\}$.

Now suppose that $F$ is a non-empty finite subset of $X\setminus\{p\}$. Then $F$ is clopen, so the function

$$f:X\to[0,1]:x\mapsto\begin{cases} 1,&\text{if }x\in F\\ 0,&\text{otherwise} \end{cases}$$

is continuous, $f[F]=\{1\}$, and $f(x)=0$ for each $x\in X\setminus F$.