Topology and Homeomorphisms Assume (X,t) is Homeomorphic to (X,t').
I think I'm correct that generally Z (being a subset of X) is dense in the topology t iff Z is dense in the topology t'.
Are there any basic counter examples to this maybe where X has two elements to negate this statement above? 

You're right, there is a two-point counterexample.
Consider $X = \{0,1\}$ with the following topologies:
$t = \{\emptyset, X, \{0\}\}$ - this is obviously a topology on $X$
$t' = \{\emptyset, X, \{1\}\}$ - also a topology on $X$
There is an obvious homeomorphism (interchanging 0 and 1) $(X,t) \rightarrow (X,t')$.
Now, take $Z = \{0\}$.
It is easy to check that $Z$ is dense in $(X,t)$, its closure being $X$. On the other hand, $Z$ is closed in $(X,t')$, so it is not dense in $(X,t')$.
