Connected Hausdorff space all whose $n$-th homology groups, for $n\ge 1$, are trivial

Question

Let $X$ be a connected Hausdorff space such that $H_n(X,\mathbb Z)=0, \forall n \ge 1$. Then does that necessarily imply that $X$ is path connected i.e. $H_0(X,\mathbb Z)=\mathbb Z$ ?

Answer 1

No : take $X$ to be the topologist's sine curve. Then $X$ us connected but not path connected.

Moreover by compactness, any $n$-simplex of $X$ either factors through the usual $\mathbb{R}^*_+ \to X$ or the point $\{*\}\to X$, and $H_n (X) = 0, n\geq 1$.