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first of all my apologies, may be my question is too stupid for a forum on Topology and Geometry, but it's something I was thinking about for a while without getting an answer : What's the actual difference between a Close set and a Complete set? I mean : from an algebraic point of view, we say that a Close set contains the limits of its successions, whereas a complete set contains the limits of its Cauchy successions. But we can show that a succession is convergent if and only if it is a Cauchy succession, so the two concepts should be equivalent, and I don't see the difference between closure and completeness. From a topological point of view, we define a close set as the complement of an open one, but at this point I don't see the connection with the completeness property. I started to taught the actual difference could be in passing from infinite dimension to finite dimension, but I am not able to see clearly why. Can anyone help me?

Sorry again if the question is not so relevant. Cheers,

Davide