TITLE: Are finite spaces a model for finite CW-complexes?
QUESTION [17 upvotes]: Are finite topological spaces (i.e. topological spaces whose underlying set is finite) a model for the homotopy theory of finite simplicial sets (= homotopy theory of finite CW-complexes) ?
Namely, is there a reasonable way to:
(1) given a finite topological space $X$, construct a finite simplicial set $nX$.
(2) given two finite topological spaces $X$ and $Y$, construct a simplicial set $Map(X,Y)$ whose geometric realisation is homotopy equivalent to the mapping space $Map(|nX|,|nY|)$ between the geometric realizations of $nX$ and $nY$.
(3) etc. (higher coherences)
Note: One can, of course, define $Map(X,Y)$ to be the derived mapping space between $nX$ and $nY$. But I'm wondering whether there's something more along the lines of "take the (finite) set of all continuous maps $X\to Y$ and then ... "

REPLY [10 votes]: Andre, the best answer to your very first question is given by Emily Clader, who 
proved that every finite simplicial complex is weak homotopy equivalent to an inverse limit of finite spaces. 
A small mistake is corrected and much further work is done in Matthew Thibault's unpublished 2013 University of Chicago thesis.
The answer to your question (1) is classical, going back to McCord as in Nardin's answer. I don't know a really good answer to (2). 
I should apologize that my book referred to by Quaochu Yuan is still unfinished.  It will be some day.  It uses the finite space of continuous maps between finite spaces to discuss homotopies in Section 2.2, but of course that is too small to realize properly.  The generalization of this to A-spaces (T_0 Alexandroff spaces) is subtle and is studied by Kukiela,
but he does not address your question (2).  
In answer to a question raised in Nardin's answer, the category of A-spaces is isomorphic to the category of posets. It was implicit in Thomason's model structure on the category of small categories that there is a similar model structure on the category of posets, and that was made explicit by Raptis.   It is Quillen equivalent to the standard model structure on simplicial sets.   That was generalized to posets with action by a discrete group G by Stephan, Zakharevich and myself. In passing, that paper somewhat streamlines the nonequivariant proof.