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?