Real Analysis(open nor closed sets)

[malcmitch20]

Hello,
I am having trouble finding an example of a set in R^2 that is neither open nor closed. I have already shown the half open interval [0,1) is neither open nor closed, but I can't seem to find any other examples. Can someone push me in the right direction? Would x^2+ y^2<1 be open nor closed? Is using the def. of a ball/neighborhood the right way to go to prove this?

 

[disregardthat]

Intuitively, a "properly behaved" subset of the R^2 is neither open nor closed if, say, it only partly contains its boundary, and otherwise contains anything infinitesimally close to its boundary. E.g. a square with its right edge removed, or a circle with a point on its boundary removed etc...

In fact, for any closed set, remove some set of points from its boundary (and boundary alone), and you will get a set which is neither open nor closed.

Otherwise it is incredibly easy to construct loads of such sets, simply take the union of two disjoint sets, one being open and one being closed.

The open disk is an open set, in fact the set of open disks form a basis for the standard topology on R^2.

 

[malcmitch20]

very informative. I didn't think of it that way. Thanks a lot!