Sets Neither Open Nor Closed

In topology, a set that is neither open nor closed does not fulfill the requirements of either category.

This phenomenon can occur in a topology where the set is neither defined as an open set nor is it the complement of one.

Therefore, the set is also not a closed set.

Note: Such sets might be hard to visualize within the familiar topology of real numbers. However, they are completely feasible in more complex topological structures. A practical example would help clarify this concept.

A Practical Example

Consider the set X={a,b,c,d} and topology T, which defines the following open sets: {b}, {a,b}, {c,d}, {b,c,d}, {a,b,c,d}, and Ø.

Let’s examine the subset {b,c} of the set X.

• The set {b,c} is not an "open set" because topology T does not explicitly define it as such.
• Similarly, {b,c} is not the complement of any open set within topology T, and thus it is not a "closed set" either.

In conclusion, according to topology T, the set {b,c} is neither open nor closed.

And so forth.