Interior and closure in non-Euclidean topology

• I
• tomkoolen
In summary, the conversation involves a question about computing the interior and closure of a set A in a topological space generated by a given basis. The space is similar to the Euclidean topology, but with different boundary points. The solution is to approach the problem in stages, considering different intervals for the given values.
tomkoolen
Hello everyone,

I was wondering if someone could assist me with the following problem:

Let T be the topology on R generated by the topological basis B:
B = {{0}, (a,b], [c,d)}
a < b </ 0
0 </ c < d

Compute the interior and closure of the set A:
A = (−3, −2] ∪ (−1, 0) ∪ (0, 1) ∪ (2, 3)

I understand that in Euclidean topology I would just include/exclude the boundary points but I don't know how to do this with a different topology, especially since I feel that this topological space is very, very similar to the Euclidean topology, I have shown that all opens in that space are open here as well. What do I do?

tomkoolen said:
a < b </ 0
0 </ c < d
Do you mean a < b ≤ 0; 0 ≤ c < d?

Yes!

The the problem must be dealt with in stages:
1. a < b < -3; 3 < c < d
2. a < -3, -3 < b <-2; 3 < c < d
3. etc. etc.

1. What is the interior of a set in non-Euclidean topology?

The interior of a set in non-Euclidean topology is the largest open set contained within the set. It includes all points within the set that do not touch the boundary of the set.

2. How is the interior different from the closure in non-Euclidean topology?

The interior and closure are complementary concepts in non-Euclidean topology. While the interior includes all points that are strictly within the set, the closure includes all points within the set as well as the boundary points.

3. Can a set have empty interior in non-Euclidean topology?

Yes, a set can have empty interior in non-Euclidean topology. This occurs when the set contains only its boundary points and no points within the set itself.

4. How is the interior of a set related to its complement in non-Euclidean topology?

The interior and complement of a set are complementary concepts in non-Euclidean topology. The interior of a set is the complement of the closure of the complement of that set.

5. What is the role of interior and closure in non-Euclidean topology in practical applications?

The concepts of interior and closure in non-Euclidean topology are important in practical applications such as computer graphics and data analysis. They help in determining the connectivity and completeness of a set, which is essential in various fields including image processing, machine learning, and network analysis.

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