Calculator
Example Data Table
| Input | Normalized Form | Closed? | Reason |
|---|---|---|---|
| [0,2] U {4} | [0, 2] ∪ {4} | Yes | All boundary points belong to the set. |
| (0,2) | (0, 2) | No | Boundary points 0 and 2 are missing. |
| (1,3] U {1} | [1, 3] | Yes | Adding 1 closes the left endpoint. |
| (-inf,3] | (-∞, 3] | Yes | Complements of this form are open rays. |
| (0,1) U (1,2) | (0, 1) ∪ (1, 2) | No | Boundary point 1 is excluded. |
Formula Used
A set A is closed iff ℝ \ A is open.
A set A is closed iff every boundary point of A lies in A, written as ∂A ⊆ A.
A set A is closed iff cl(A) = A. For finite unions on the real line, closure adds any missing boundary endpoints.
How to Use This Calculator
- Enter a real-line set such as
[0,2] U {4} U (5,7]. - Choose a proof style, or leave it on Automatic.
- Set the graph window and sample density.
- Click Analyze Set to place the result above the form.
- Review the normalized set, interior, boundary, closure, and complement.
- Use the export buttons to save the result as CSV or PDF.
Scope Notes
This page analyzes finite unions of intervals and finite point sets on the real line. That scope keeps the proof logic rigorous, visual, and fast for common topology exercises.
FAQs
1) What inputs does this calculator support?
It supports finite unions of intervals, finite point sets, single numbers, ℝ, and ∅ on the real line.
2) Why can a set be closed but not open?
Closed sets contain their boundary points. Open sets exclude boundary points. A set may satisfy only one condition.
3) Can unions turn a non-closed piece into a closed set?
Yes. For example, (1,3] ∪ {1} becomes [1,3], which contains both finite boundary endpoints.
4) Does this tool work in higher dimensions?
No. This version is built for subsets of ℝ, where interval notation and boundary checks stay exact.
5) What does the boundary test mean?
It checks whether every boundary point belongs to the set. Missing even one boundary point makes the set non-closed.
6) Why does closure add endpoints?
Closure adds all limit points. On the real line, that often means attaching excluded finite boundary endpoints.
7) What do the CSV and PDF exports include?
They include the normalized set, classification, interior, boundary, closure, complement, and a concise proof summary.
8) What should I do if parsing fails?
Check brackets, commas, and union symbols. Write expressions like [0,2] U {4} U (5,7].