In essence, this is the same problem as in 
"The generalization of Brouwer's fixed point theorem?".
But now I am determined to be careful. The main question is 
the following:

Is there any generalization of Brouwer's fixed point theorem 
in terms of general topology? (That is, without triangulations, 
or vector spaces, or anything else).

The question is a bit vague, but, I hope, it admits a
precise answer. Now, I am trying my best to propose a candidate.  
 
Let $X$ be a contractible locally contractible
Hausdorff second countable  compact topological space. 
Let $f\colon X\to X$ be a continuous map. 
Has then $f$ a fixed point?