Azeotropes and separation by distillation?
You seem to have the right idea.

To elaborate, take the specific example of the positive azeotrope of (roughly) 96% ethanol and 4% water. If you begin with a mixture containing less than 96% ethanol, distillation will result in a distillate more abundant in ethanol and nearer to the azeotropic composition. That implies the residue will necessarily be proportionally lower in ethanol. If you continuously repeat the distillation procedure using the remaining pot residue, it will approach 100% water. On the other hand, if you begin with a concentration above 96% ethanol, the resulting distillate will actually be lower in ethanol (and therefore nearer to the azeotropic point), while the residue will actually become more concentrated in ethanol. Hence, in theory, with many repeated distillations you can approach purity in the residue when dealing with positive azeotropes, and the ultimate composition of the residue will depend on which side of the curve you're on to start with.

When dealing with negative azeotropes, the situation is analogous but the particulars are basically reversed. Take the example of hydrofluoric acid and water, which form an azeotrope at approximately 37% HF. If you initially begin distilling a mixture comprising less than 37% HF, you'll obtain a distillate more dilute in HF, while the residue will contain more HF and be nearer to the azeotrope. If you were to collect the distillate and repeat the distillation, you would again end up with an even more dilute distillate, with the concentration of the distillate approaching 100% pure water. If you begin distilling a mixture containing more than 37% HF, then the residue will end up being more dilute as it approaches the azeotrope, while the distillate will actually be richer in HF. If you repeat this process many times over, you can eventually obtain a distillate approaching 100% pure HF (in theory).

So, in summary, you're correct to suggest that one of the components can be isolated with something approaching complete purity from an azeotropic mixture, assuming a sufficient volume to enable an adequate number of simple distillation cycles or theoretical plates in fractional distillation. Of course, the problem is that you really have no choice in determining which of the two components you can isolate, as that depends entirely on the starting composition of the mixture. In practice, you can often separate azeotropeic mixtures by modifying the distillation process in various ways.