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Let $X$ be a compact Haussdorf topological space with the following property:

For every continuous function $f:X\to \mathbb{C}$ the image $f(X)$ is a path connected subset of $\mathbb{C}$.

Is such a space $X$ necessarilly a path connected space?

This question is inspired by (and realated to) the following post:

Unital $C^{*}$ algebras whose all elements have path connected spectrum

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    $\begingroup$ I think the extended long line $\bar L$ (add one point at each end) is a counterexample. Every map $\bar L \to \mathbf R$ (or to $\mathbf C = \mathbf R^2$) is eventually constant so the image is path connected, but $\bar L$ itself is too long to be path connected. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Apr 25, 2020 at 3:32
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    $\begingroup$ The one-point compactification of $\omega_1\times [0,1)$ with the order topology is indeed a counterexample. This question might be more interesting with an extra hypothesis on $X$, e.g. sequential, first countable, or metrizable. $\endgroup$
    – Jeremy Brazas
    Commented Apr 25, 2020 at 3:38
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    $\begingroup$ @JeremyBrazas What about metrizable counterexamples? $\endgroup$
    – Taras Banakh
    Commented May 2, 2020 at 7:33
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    $\begingroup$ @TarasBanakh That is the question I would ask...As stated, the question is answered. I cannot tell if the OP is satisfied or wants other cases addressed. My comment is meant to encourage the OP to revise the question. I will consider thinking about it more when I know what the revised question really is. $\endgroup$
    – Jeremy Brazas
    Commented May 2, 2020 at 13:07

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