Types of points in metric spaces

[pob1212]

Hi,

I'm reading Baby Rudin and have a quick question regarding topology.

Given a nonempty subset E of a metric space X, is it true that the only points in E are either isolated points or limit points? (b/c all interior points are by definition limit points, but not all limit points are interior points)

Does this exhaust every kind of point in E?

Thanks,
pob

 

[rasmhop]

Yes every point of E is either an isolated point or a limit point of E.

The proof is not complicated and you should consider it an exercise. It is the kind of statement you should be comfortable proving after reading this section of Baby Rudin.