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It's not a common false belief, IMO. On the first day you learn about open and closed sets, you learn that the for any metric space S, the set S is open in S a
by SamReidHughes 5y ago
It's not a common false belief, IMO. On the first day you learn about open and closed sets, you learn that the for any metric space S, the set S is open in S and the empty set is open in S, and they're closed as well. (Students typically get introduced the concepts in the context of metric spaces.)
Note that sets are always closed or open relative to some other specified (often implied) set S.
In some courses, a closed set (in S) is defined to be a set which is the complement S-T for some open set T. In others, a closed set (in S) is defined to be a set which contains all its limit points (in S). And then whichever isn't the definition gets proven as a theorem.
- Someone 5y agoI think the first _and_only_ day many people learn about open and closed sets is when they learn about the specific example of open and closed intervals in ℝ. That happens early in high school. The misconception is about not knowing that the generalization to arbitrary sets has some unexpected properties. (Half-open intervals are neither open nor closed, by the way)
- SamReidHughes 5y agoI never learned about open and closed "sets" in R, in high school, only intervals.
- Nitrolo 5y agoI guess all intervals are sets (they are a collection of numbers after all) but not all sets are intervals. Any mathematicians here to tell me if I just said something stupid?
- SamReidHughes 5y agoNo, that's correct.
- BlueTemplar 5y agoIndeed, sets come later, around university, and I would assume that most university fields that use mathematics do NOT need to care about some sets like R (and C ?) having the clopen property ?
- recursive 5y agoSo can a set that's not an interval be open or closed?