Why is Completeness not a Topological Property? I am trying to answer the question:
Show why completeness is not a topological property.

My answer:  $\mathbb{R}$ and the set $(0,1)$ are homeomorphic, but $\mathbb{R}$ is complete while $(0,1)$ is not.

My question to you all:  Does this answer the question?  I feel like I am not quite seeing what is going on with completeness and why it is not a topological property. Can someone give me another example?
Another way to see it is:  One can metrize the space $(0,1)$ in (at least) two different ways that generate the same topology.  One such metric is the ordinary Euclidean metric on $(0,1)$; a second metric is what you get when you pull the Euclidean metric on $\mathbb{R}$ back to $(0,1)$ via a homeomorphism.

In one of those metrics, $(0,1)$ is complete, but in the other it is not -- again, despite the fact that the two metrics generate the same topology. That shows that completeness depends not on the topology per se but rather on the metric one associates with the topological space.
The completenes property is dependent upon the metric. If you change the metric, the same Cauchy sequences which once converged may not under your new metric.  Conceptually, size or distance are not properties preserved under homeomorphism.  Topology is concerned with structure - limit point relations on the set - continuity, connectedness, compactness.