# 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.