I was asked whether or not the following is true:
Let $Y$ be a topological space, $X \subset Y$ a subspace, and $f: X \hookrightarrow Y$ the inclusion map. Then the induced map on homology $f_{*}:H_n(X) \to H_n(Y)$ is always injective.
This is of course false. Take $S^1$ embedded in $S^2$, par exemple. Then $H_1(S^1) = \mathbb{Z}$ while $H_1(S^2) = 0$.
But this got me thinking: What if we restrict ourselves to the $0$th homology?
If $f: X \hookrightarrow Y$ is the inclusion map, is the induced map $H_0(X) \to H_0(Y)$ on $0$th homology always injective?
Here a counterexample would be taking two distinct points $x,y \in S^1$. Then $H_0(\{x,y\}) = \mathbb{Z} \oplus \mathbb{Z}$ while $H_0(S^1) = \mathbb{Z}$.
But if we restrict ourselves even more:
If $X$ is a connected subspace of $Y$, and $f: X \hookrightarrow Y$ is the inclusion map, is the induced map $H_0(X) \to H_0(Y)$ on $0$th homology always injective?
I have been unable to find a counterexample, but am also unsure as to how one would prove this.
All help would be much appreciated.
Best Answer
This is not true, take the topologists sine curve embedded in $\mathbb R^2$ for example. It is connected but not path connected, it has two path components so its $0-th$ homology is $\mathbb Z \oplus \mathbb Z$ and of course the $0-th$ homology of $\mathbb R^2$ is $\mathbb Z$