How to determine the family of bounded functions from an infinite Fort space to $[0,1]$?

Question

Definition: Let $X$ be a topological space and $b\in X$. We call $X$ a Fort space (with particular point $b$), when $X$ has topology $\{A\subseteq X: b \not\in A \; \text{or} \; X\setminus A\; \text{is finite}\} $.

It is clear that a Fort space $X$ with particular point $b$, $X\setminus b$ is discrete and $X$ is one-point compactification of $X\setminus b$. So, a Fort space is simply ''Alexandroff Compactification of a Discrete Space''.

Now, let $X$ be an infinite Fort space with particular point $b$. I want to determine the family of all bounded functions from $X$ to $[0,1]$.

Actually, I don't know where to start and what to search for... So, any help is definitely appreciated.

Answer 1

Since any function from $X$ into $[0,1]$ is bounded, I suppose you want to characterize the continuous functions from $X$ into $[0,1]$.

If you choose any sequence $a=\langle a_n : n \in \omega \rangle$ in $X$ and any convergent sequence $r=\langle r_n : n \in \omega \rangle$ in $[0,1]$ then the function $f_{a,r}:X \to [0,1]$, defined by $f_{a,r}(a_n)=r_n$ and $f_{a,r}(x)=\lim_{n \to \infty}r_n$ for $x \notin \{a_n : n \in \omega\}$, is continuous. On the other hand, since $[0,1]$ is first countable, it is not hard to show that any continuous function from $X$ into $[0,1]$ is of this form.