Calculator
Formula used
- A is open β A β π―
- A is closed β U \ A is open
- Int(A) = β{ O β π― : O β A }
- Cl(A) = U \ Int(U \ A)
- Bd(A) = Cl(A) \ Int(A)
How to use this calculator
- Type the elements of U separated by commas.
- Enter the topology π― as a list of sets, separated by semicolons.
- Provide a subset A. Add B if you want set operations.
- Optionally enter a point x to test interior membership.
- Press Submit to display results above the form.
- Use Download CSV or Download PDF to export the results table.
Example data table
| Universe U | Topology π― | Subset A | Interior Int(A) | Closure Cl(A) |
|---|---|---|---|---|
| {a, b, c, d} | β , {a}, {a, b}, {a, b, c, d} | {a, b} | {a, b} | {a, b, c, d} |
| {1, 2, 3} | β , {1}, {1, 2}, {1, 2, 3} | {2} | β | {1, 2, 3} |
Professional notes and data
Finite topology inputs and reliability
This calculator is designed for finite universes, where every set can be listed explicitly. With |U| = n, the power set has 2^n subsets, but a topology is usually much smaller. Many classroom examples use n = 3 or n = 4, where 2^n equals 8 or 16, making manual checking possible but slow.
Topology axiom checks on your list
For the sets you enter, the tool validates three essentials: presence of β and U, closure under unions, and closure under intersections. On finite inputs, pairwise checks provide a practical consistency test. If your topology has m listed open sets, the pairwise scan touches up to mΒ² combinations.
Core outputs and what they quantify
The key derived sets are interior, closure, and boundary. Interior reports the largest open part of A, closure reports all points that cannot be separated from A using open sets, and boundary measures the βedgeβ as Cl(A) \ Int(A). The Plotly chart summarizes their element counts instantly.
Operational data for set operations
When you provide an optional B, the tool computes A βͺ B, A β© B, A \ B, and symmetric difference A β³ B. Each result is reduced to unique elements and sorted, ensuring consistent output for exports and comparisons across runs.
Exports for reporting and auditing
CSV export captures the results table as keyβvalue rows for spreadsheets and notebooks. PDF export packages the same table into a clean page for sharing. In typical coursework reviews, a single PDF summary can replace multiple screenshots and reduces transcription errors from manual copying.
Recommended input sizes and performance
For best responsiveness in a browser, keep |U| under 30 and the listed topology size m under 150. That still supports thousands of union/intersection comparisons while remaining usable on laptops. If you need larger experiments, split the study into smaller universes and compare summaries using the CSV exports.
FAQs
What does βA is openβ mean here?
It means your subset A appears in the topology list π― exactly. If A is not listed, it is treated as not open in that topology.
Why can A be closed even if it is not open?
A is closed when its complement U\A is open. In many topologies, only a few sets are open, so complements can determine closedness.
How is the interior computed?
Interior is the union of every open set O in π― that is contained in A. The result is the largest open subset you can guarantee inside A.
How is closure computed?
Closure is computed as U \ Int(U \ A). This definition avoids neighborhood searches and works well for finite inputs where complements and unions are explicit.
What does the boundary represent?
Boundary equals Cl(A) \ Int(A). These elements are neither safely inside A nor safely outside A, based on the open sets you provided.
Why might the topology checker warn me?
Warnings appear when β or U is missing, or when a pairwise union/intersection of your listed open sets is not listed. That indicates your input list is not a valid topology.