Assume that $X$ is a path-wise connected Hausdorff space, and assume that its fundamental group is non-trivial. Does it always exist a simple curve in $X$ which is non-null-homotopic?


Such curve does exist if we further assume that $X$ possesses a universal cover.



Existence of a non-null-homotopic simple closed curve
No. The harmonic archipelago is a compact $2$-dimensional counterexample, embedded in $\mathbb R^3$.


Let $S\_n$ denote the planar circle with center on the $x$-axis, and whose intersection with the $x$-axis is 
$$
\left\{\Big(\frac{1}{n+1},0\Big), \Big(\frac{1}{n},0\Big)\right\}.
$$ Take the closure of the union of the circles. Now make each circle the base of a cone, raised in the $z$-direction, of height $1$, and take the closure of the union of the cones. Every simple closed curve is inessential, but the space is not simply connected. (A curve running once around each base circle is essential).


