General Topology – How to Determine the Family of Bounded Functions from an Infinite Fortune Space to $[0,1]$?

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

It is clear that a strong space $ X $ with a particular point $ b $, $ X setminus b $ is discreet and $ X $ is the compactification at a point of $ X setminus b $. So, a strong space is just "#"; Alexandroff's compactification of a discreet space "."

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

In fact, I do not know where to start or what to look for … So any help is really appreciated.