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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?

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I do not have the answer but these matters are discussed extensively in Bing "The elusive fixed point property". Amer. Math. Monthly 76 1969 119–132. jstor.org/pss/2317258. –  Igor Belegradek Mar 31 '11 at 13:36
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As I remarked in my answer to your previous question, this seems to be open. –  Daniel Litt Apr 1 '11 at 16:53
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