My last exposure to topology was almost a year ago and that was at an undergraduate level - Nonetheless, I have grown to be comfortable with topology.
I am currently doing a research with a visiting scholar and would like to know if there exists a theorem or lemma that would enable us to determine if
1) given a respective subset S of space in $\mathbb{R}^{2}$, does there exists a space T in $\mathbb{R}^{2}$ that is topologically equivalent to this space?
2) Now, suppose that there exists randomly assigned fixed number of points along the boundary of the space S. The randomly assigned fixed number of points along the boundary of the space S is a random variable with a corresponding probability distribution.
Is there a lemma or theorem that would allow us to determine the probabilities of points under a map?
Any help is appreciated.
Tangent: Can this same question be posted on math overflow?