It's well-known that every homology theory satisfying Eilenberg-Steenrod axioms is isomorphic to singular homology. I tried to perform some homology calculations directly from axioms but couldn't do even a simple task: I can't prove that $H_0$ of a (linearly) connected space is isomorphic to $H_0$ of a point. It's not hard to prove (by considering morphism of exact sequences of pairs $(X, *) \to (*, *)$) that the map $H_0(*) \to H_0(X)$ induced by inclusion of a point is a monomorphism, and $X$ being linearly connected implies that any such inclusion induces the same map (they are homotopic). I see no way to use this fact to prove surjectivity.
Calculation of $H_0$ looks by an order of magnitude simpler than proving the uniqueness of homology theory, so I hope there is a quick and elementary argument for it. Could someone help me?