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1. ### Not so open minded open sets

I can see how a set could be neither open nor closed, but I cannot invision something concrete to see how a set could be both open and closed, save defining some set to not contain its boundary points. I guess I'm a bit new to this game, because things such as the above still seem way to...
2. ### Not so open minded open sets

Actually, as to the definition of closed sets: I could understand if it required the sup and inf to be included in order to be closed, but that's not how the definition reads :confused:
3. ### Not so open minded open sets

*Definitions* An open set: A set is open if every point of the set lies in an open interval entirely contained in the interval. A closed set: A set is said to be closed if it contains all its limit points. I'm ok with the definition of open sets, but I'm still working on really...
4. ### Not so open minded open sets

The union/intersection/whatever neurocomp2003 meant to say about [-1, 1) and (-1, 1] is something I can understand. Correct me if I'm wrong, but the intersection of the two would be open and the union would be closed, right? I also can understand why the set [-1, 1) is neither open or...
5. ### Not so open minded open sets

Wouldn't the intersection of all the A_n's be zero? So, if that's the case then I take it that zero is not considered an open set? Zero does not seem like it should be an open set.
6. ### Not so open minded open sets

This is not a specific homework question so much as it is a general conceptual question. My analysis book includes a theorem that states: 1. The union of any number of open sets is an open set. 2. The intersection of a finite number of open sets is an open set. I follow the proof of...