Relative homology and path connected space 2

In this post (Relative homology and path connected space) it is proved that if $X$ is a path connected space and $Y\subset X$ is not empty, then $H_0(X,Y)\simeq0$. I was wondering if the reciprocal is true; more exactly, suppose for all subset $Y$, $Y\not=\emptyset$, of a topological space $X$, $H_0(X,Y)\simeq0$ holds. Can we conclude that $X$ is path connected?

Solution 1:

$H_n(X,x_0)\simeq \widetilde{H_n}(X)$ for any point $x_0\in X$: Homology relative to a point

And thus $H_0(X,x_0)\simeq 0$ for some (any) point $x_0\in X$ is equivalent to $\widetilde{H_0}(X)\simeq 0$ which is equivalent to $X$ being path connected: The zeroth homology group corresponds to path components