An open set is a subset where each point has a neighborhood contained in the set; in metric spaces this is witnessed by a ball around every point.

How does this help with continuity?

Continuity can be detected by whether the preimage of every open set is open, a condition that the Open Sets skill helps verify using solver proofs.

The skill leverages Z3-based proofs via scripts like preimage_open, ball_contained, and epsilon_delta_bound (e.g., via z3_solve.py).

open-sets

npx machina-cli add skill parcadei/Continuous-Claude-v3/open-sets --openclaw

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SKILL.md

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Open Sets

When to Use

Use this skill when working on open-sets problems in topology.

Decision Tree

Is f: X -> Y continuous?
- For metric spaces: x_n -> x implies f(x_n) -> f(x)?
- For general spaces: f^(-1)(open) = open?
- For products: Check each coordinate function
- z3_solve.py prove "preimage_open"
Open Set Verification
- For metric spaces: for all x in U, exists epsilon > 0 with B(x,epsilon) subset U
- z3_solve.py prove "ball_contained" with epsilon witnesses
Topological Properties
- Interior: int(A) = largest open subset of A
- Closure: cl(A) = smallest closed superset of A
- Boundary: bd(A) = cl(A) \ int(A)
Continuity Tests
- Epsilon-delta: for all epsilon > 0, exists delta > 0: d(x,a) < delta implies d(f(x),f(a)) < epsilon
- z3_solve.py prove "epsilon_delta_bound"

Tool Commands

Z3_Preimage_Open

uv run python -m runtime.harness scripts/z3_solve.py prove "preimage_open"

Z3_Epsilon_Delta

uv run python -m runtime.harness scripts/z3_solve.py prove "ForAll(eps, Exists(delta, d(x,a) < delta implies d(f(x),f(a)) < eps))"

Z3_Ball_Contained

uv run python -m runtime.harness scripts/z3_solve.py prove "ball_contained"

Key Techniques

From indexed textbooks:

[Introduction to Topological Manifolds... (Z-Library)] Show that every local homeomorphism is an open map. Show that every homeomorphism is a local homeomorphism. Show that a bijective continuous open map is a homeomorphism.
[Introduction to Topological Manifolds... (Z-Library)] The key motivation behind the denition of this new kind of space is the open set criterion for continuity (Lemma A. Appendix), which shows that continuous functions between metric spaces can be detected knowing only the open sets. Motivated by this observation, we make the following denition.
[Introduction to Topological Manifolds... (Z-Library)] Suppose X is a set, and B is any collection of subsets of X whose union equals X. Let T be the collection of all unions of nite inter- sections of elements of B. Note that the empty set is the union of the empty collection of sets.
[Introduction to Topological Manifolds... (Z-Library)] The product topology is “associative” in the sense that the three prod- uct topologies X1 × X2 × X3, (X1 × X2) × X3, and X1 × (X2 × X3) on the set X1 × X2 × X3 are all equal. For any i and any points xj ∈ Xj, j = i, the map fi : Xi → X1 × · · × Xn given by fi(x) = (x1, . If for each i, Bi is a basis for the topology of Xi, then the set {B1 × · · · × Bn : Bi ∈ Bi} is a basis for the product topology on X1 × · · · × Xn.
[Introduction to Topological Manifolds... (Z-Library)] Here are some examples of closed subsets of familiar topological spaces. Any closed interval [a, b] ⊂ R is a closed set, as are the half-innite closed intervals [a, ∞) and (−∞, b]. Every subset of a discrete space is closed.

Cognitive Tools Reference

See .claude/skills/math-mode/SKILL.md for full tool documentation.

Source

git clone https://github.com/parcadei/Continuous-Claude-v3/blob/main/.claude/skills/math/topology/open-sets/SKILL.mdView on GitHub

Overview

This skill focuses on solving open-set problems in topology by testing openness via preimages, ball containment in metric spaces, and standard topological properties. It also covers continuity tests and how to formalize proofs with Z3-based checks.

How This Skill Works

Apply the decision tree: verify if f^{-1}(open) is open for continuity, confirm ball containment around each point for open sets in metric spaces, and compute interior, closure, and boundary as needed. Use the provided Z3 tool commands (preimage_open, ball_contained, epsilon_delta_bound) to automate proofs.

When to Use It

You need to verify a function’s openness/continuity by checking preimages of open sets.
You’re testing whether a subset is open in a metric or topological space using ball criteria.
You want to determine interior, closure, or boundary of a set to understand its topology.
You are checking continuity via epsilon-delta in metric spaces and seeking formal witnesses.
You’re working with product or coordinate spaces and need to verify openness across coordinates.

Quick Start

Step 1: Identify the map f: X -> Y and the topologies on X and Y.
Step 2: Apply the decision tree to test openness: check preimage_open, then ball_contained if needed.
Step 3: Use the solver commands (preimage_open / ball_contained / epsilon_delta_bound) to obtain proofs.

Best Practices

Follow the decision tree strictly: Is f: X -> Y continuous? then verify open preimages.
For metric spaces, prove openness by showing every x in U has a ball B(x,epsilon) contained in U with explicit epsilon.
Use z3_solve.py proofs with preimage_open, ball_contained, and epsilon_delta_bound to generate and check witnesses.
Keep interior, closure, and boundary relations in mind: int(A), cl(A), and bd(A) guide open-set reasoning.
Document each open-set argument with clear justifications or solver outputs to aid reproducibility.

Example Use Cases

Determine if a function between metric spaces is open by proving the preimage of each open set is open.
Show a subset of R is open using ball containment around each point with a suitable epsilon.
Compute int(A), cl(A), and bd(A) for a given subset A in a familiar space like R or R^n.
Use z3_solve.py to prove preimage_open for a proposed continuous function.
Validate continuity of a function via an epsilon-delta bound and verify solver outputs.

Frequently Asked Questions

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