Calculator
Build the set and test the proof
Example data table
| Case | Submitted set | Sample point | Witness or issue | Verdict |
|---|---|---|---|---|
| 1 | (-2, 1) ∪ [1, 4) | 0 | ε = 1 works at x = 0 | Not open because 1 is an included finite boundary |
| 2 | (0, 2) ∪ [2, 5) | 2 | ε = 1 shows interior coverage at the joining point | Open after merging to (0, 5) |
| 3 | (-∞, 3) ∪ (4, ∞) | 5 | ε = 1/2 keeps the ball inside (4, ∞) | Open |
Formula used
Open-set criterion in a metric space:
A set U is open when every point x ∈ U has some ε > 0 with B(x, ε) ⊆ U.
For interval unions on the real line, the calculator merges overlapping or truly touching pieces and then inspects the finite outer endpoints of each merged component.
If a merged component is (a, b) and x lies inside it, a convenient safe witness is ε = ½ min(x - a, b - x).
If a finite boundary point is included, that boundary point becomes a counterexample because every ε-neighborhood crosses the complement.
How to use this calculator
- Enter a set label, sample point, and optional study note.
- Add one or more interval rows describing the union you want to test.
- Choose open, closed, or infinite boundaries for each endpoint.
- Submit the form to merge the intervals and analyze openness.
- Read the proof summary, witness note, and any boundary counterexample.
- Use the graph plus CSV and PDF exports for revision or classwork.
FAQs
1) What does this calculator prove?
It checks openness for finite unions of intervals on the real line. The result explains merged components, a witness radius for the sample point, and any included finite boundary that blocks openness.
2) Why are intervals merged first?
A proof should study the actual union, not only the raw rows. Merging reveals the true connected pieces and can show that touching intervals combine into one larger component.
3) Can touching intervals still produce an open set?
Yes. For example, (0, 2) ∪ [2, 5) becomes (0, 5). The joining point is interior because nearby points on both sides remain inside the union.
4) Why does an included finite boundary fail?
At an included finite boundary, every ε-ball contains points outside the set on the missing side. That means no positive witness radius keeps the whole neighborhood inside the set.
5) What is the witness radius?
It is a convenient positive ε guaranteed to stay inside the containing merged component. This page usually returns half the distance from the sample point to the nearest boundary.
6) Does this handle every subset of ℝ?
No. This version is designed for finite unions of intervals. Sets such as discrete point collections, Cantor-like constructions, or arbitrary formulas need different input models.
7) Can I type fractions?
Yes. Enter integers, decimals, or simple fractions like 3/2. Infinite endpoints should be selected from the boundary menus instead of typed into the value boxes.
8) What do the CSV and PDF downloads include?
They include the main verdict, sample-point evidence, and merged component table. That makes the page handy for homework checking, worksheets, and classroom revision notes.