About general Homology with negative index Let $0 > n\in \mathbb{Z}$. I know, that in Singular Homology $ H_n(X)=0 $ for a toplogical space $ X \neq \emptyset $. If we now use a general Homology which satisfies the Eilenberg–Steenrod axioms does this property also hold? If yes, how to prove? And if no, is there an example of a Homology theory which is non zero for negative $n$?
Furthermore if it is not zero, how does this affect the reduced homology?
 A: No.
The "classical" Eilenberg-Steenrod axioms are homotopy invariance, exactness, excision and dimension. These describe ordinary homology theories. It is well-known that for finite CW-pairs the homology groups $H_n(X,A)$ are (up to natural isomorphism) uniquely determined by the coefficient group $G = H_0(*)$, where $*$ is a one-point space. In fact, they agree with the singular homology groups of $(X,A)$ with coefficients in $G$.
In particular $H_n(X,A) = 0$ for $n < 0$.
Beyond finite CW-pairs things are more sophisticated. In

I.M. James and J.H.C. Whitehead, "Homology with zero coefficients", The Quarterly Journal of Mathematics, Volume 9, Issue 1, 1958, Pages 317–320

one can find examples of non-trivial ordinary homology theories with zero coefficient group. There are infinite CW-complexes $X$ with nonvanishing homology groups. Making a dimension shift ($H'_n = H_{n+k}$ for some $k \in \mathbb N$) we can achieve that $H'_n(X) \ne 0$ for negative $n$. The theory $H'_*$ has coefficient group zero, if you do not like that consider $H'_* \oplus H^{sing}_*$, where $H^{sing}_*$ is singular homology with $\mathbb Z$-coefficients.
