Many times one reads that ``this is a topological property, while that is a metric property.''. What does this mean?
Well, in general, perhaps one should say: nothing.
That is, consider standard English. We may talk about shape, and we may talk about color, and we may even use both in describing the same object. But in general, they are simply different descriptive domains.
However, this isn't entirely satisfactory. For many of the mathematical structures we may examine, the two concepts are related. For instance, if we define a metric on a space, that will induce a topology on the space, which in the abstruse way of mathematicians, we generally call the Induced Metric Topology.
So how can we get a ``feel'' for the relationship between topological properties and metric ones? One inroads might be the relationship between Closed Sets (topological) and Complete Sets (metric).