Return to Question

Let $X$ be a contractible compact locally connected topological space (Hausdorff and second countable). Let $f\colon X\to X$ be a continuous map. Then (I suppose) $f$ has a fixed point. Personally, I cannot think of a better generalization of Brouwer's fixed point theorem, but is it true?

Let $X$ be a contractible compact [edit: locally connected] topological space (Hausdorff and second countable). Let $f\colon X\to X$ be a continuous map. Then (I suppose) $f$ has a fixed point. Personally, I cannot think of a better generalization of Brouwer's fixed point theorem, but is it true?

Let $X$ be a contractible compact topological space (Hausdorff and second countable). Let $f\colon X\to X$ be a continuous map. Then (I suppose) $f$ has a fixed point. Personally, I cannot think of a better generalization of Brouwer's fixed point theorem, but is it true?

Let $X$ be a contractible compact locally connected topological space (Hausdorff and second countable). Let $f\colon X\to X$ be a continuous map. Then (I suppose) $f$ has a fixed point. Personally, I cannot think of a better generalization of Brouwer's fixed point theorem, but is it true?