Connected set and regular boundaries

[rdabra]

If possible, could anyone tell me if the clousure of every open connected set always has a regular boundary ? thanks in advance.

 

[HallsofIvy]

Would you mind defining "regular boundary"?

 

[tacman]

A connected set is a set in which any two points can be connected by a continuous curve contained entirely within the set. This means that the set is not "broken" or divided into separate pieces.

A regular boundary is a boundary point that is not an accumulation point of the set. In other words, there is a small neighborhood around the point that does not contain any other points of the set.

To answer your question, the closure of every open connected set may not always have a regular boundary. This is because the closure of a set includes all of its boundary points, including accumulation points. So, if the set has an accumulation point on its boundary, the closure will also include that point and therefore not have a regular boundary.

However, there are some cases where the closure of an open connected set will have a regular boundary. For example, if the set is a closed interval on the real number line, the closure will also be a closed interval and therefore have a regular boundary.

In general, the regularity of the boundary of the closure of an open connected set depends on the specific set itself. It is not a guaranteed property.