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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?

M. Rahmat
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  • No, we cannot. I suggest you carefully read the answer of the other post you link, as it already answers this question. – Thorgott Jan 08 '22 at 14:50
  • @Thorgott care to elaborate why "no"? By taking $Y$ to be a single point the other answer implies the same thing, that a single point has to meet each path component, and so the space is path connected. – freakish Jan 08 '22 at 20:06
  • Oh, I apologize, I read "a" instead of "all" for the subsets $Y$ (which would be the actual converse to the statement, unlike what it actually says). I agree then, of course, that, yes, we can. – Thorgott Jan 08 '22 at 20:59

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$H_n(X,x_0)\simeq \widetilde{H_n}(X)$ 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

freakish
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  • Thanks. So the answer is YES contrary to what Thorgott says. Correct? – M. Rahmat Jan 08 '22 at 17:22
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    @M.Rahmat I think that the answer is YES. We don't need to look at all subsets $Y$, it is enough when $H_0(X,x_0)\simeq 0$ for some point $x_0$. I don't see any issue with my answer. And I don't really understand Thorgott's comment. The answer to the other question actually says the same thing via different method. – freakish Jan 08 '22 at 20:01