Walter Rudin define a closed set as:
2.18 (d) A subset E of a metric space is closed if every limit point of E is a point of E.
I don't see in this definition nothing about the existence of a limit point of such set E. It's like the definition of a compact set, I don't see nothing about the existence of a open cover of such set, but I know that there is at least one open cover, its own metric space.
For me, the definition of a closed set doesn't guarantee the existence of a limit point, ¿am I right or not? and ¿Why?