# Math Help - About a short exact sequence of relative homology groups.

1. ## About a short exact sequence of relative homology groups.

Let $x$ be a point in $X$. My question is about the following short exact sequence:

$0 \to H_1(X) \to H_1(X,\{x\}) \to H_0(\{x\}) \: \xrightarrow{i_\ast} \: H_0(X)\to H(X,\{x\})\to 0,$

where $i_* : H_0 (\{ x \} ) \to H_0(X)$ is the homomorphism induced by the obvious inclusion. I am told that $i_*$ is injective. Although this does make some sense intuitively, I can't really prove it. Can someone show me why it is?

2. ## Re: About a short exact sequence of relative homology groups.

Cryptic as it may sound, it holds because all definitions become trivially true for inclusion maps.