Real Analysis(open nor closed sets)

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SUMMARY

This discussion focuses on identifying sets in R² that are neither open nor closed. The user successfully identifies the half-open interval [0,1) as an example and explores the concept of sets like x² + y² < 1, questioning their classification. Key insights include that removing points from the boundary of a closed set results in a set that is neither open nor closed, and that the union of disjoint open and closed sets also forms such sets. The open disk is highlighted as an example of an open set, which serves as a basis for the standard topology on R².

PREREQUISITES
  • Understanding of open and closed sets in topology
  • Familiarity with R² and its geometric properties
  • Knowledge of basic set operations, including unions and intersections
  • Concept of neighborhoods and balls in metric spaces
NEXT STEPS
  • Study the properties of open and closed sets in topology
  • Learn about the concept of basis for a topology, specifically in R²
  • Explore examples of sets that are neither open nor closed in various contexts
  • Investigate the implications of removing boundary points from closed sets
USEFUL FOR

Mathematicians, students of real analysis, and anyone interested in the properties of sets in topology will benefit from this discussion.

malcmitch20
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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?
 
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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.
 
very informative. I didn't think of it that way. Thanks a lot!
 

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