Agent Skills
› parcadei/Continuous-Claude-v3
› compactness
compactness
GitHub提供拓扑学中紧致性问题的解决策略，涵盖判定方法（如海涅-博雷尔定理）、乘积空间性质及连续映射后果。结合Z3和SymPy工具辅助证明与计算。
触发场景
需要判断集合是否紧致
验证序列收敛性
计算紧集上的函数极值
安装
npx skills add parcadei/Continuous-Claude-v3 --skill compactness -g -y
SKILL.md
Frontmatter
{
"name": "compactness",
"description": "Problem-solving strategies for compactness in topology",
"allowed-tools": [
"Bash",
"Read"
]
}
Compactness
When to Use
Use this skill when working on compactness problems in topology.
Decision Tree
-
Is X compact?
- If X subset R^n: Is X closed AND bounded? (Heine-Borel)
- If X is metric: Does every sequence have convergent subsequence?
- General: Does every open cover have finite subcover?
z3_solve.py prove "bounded_and_closed"
-
Compactness Tests
- Heine-Borel (R^n): closed + bounded = compact
- Sequential: every sequence has convergent subsequence
sympy_compute.py limit "a_n" --var nto check convergence
-
Product Spaces
- Tychonoff: product of compact spaces is compact
- Finite products preserve compactness directly
-
Consequences of Compactness
- Continuous image of compact is compact
- Continuous real function on compact attains max/min
sympy_compute.py maximum "f(x)" --var x --domain "[a,b]"
Tool Commands
Z3_Bounded_Closed
uv run python -m runtime.harness scripts/z3_solve.py prove "bounded_and_closed"
Sympy_Limit
uv run python -m runtime.harness scripts/sympy_compute.py limit "a_n" --var n --at oo
Sympy_Maximum
uv run python -m runtime.harness scripts/sympy_compute.py maximum "f(x)" --var x --domain "[a,b]"
Key Techniques
From indexed textbooks:
- [Topology (Munkres, James Raymond) (Z-Library)] CompactSpaces163 164ConnectednessandCompactnessCh. Itisnotasnaturalorintuitiveastheformer;somefamiliaritywithitisneededbeforeitsusefulnessbecomesapparent. AcollectionAofsubsetsofaspaceXissaidtocoverX,ortobeacoveringofX,iftheunionoftheelementsofAisequaltoX.
- [Real Analysis (Halsey L. Royden, Patr... (Z-Library)] If X contains more than one point, show that the only possible extreme points of B have norm 1. If X = Lp[a, b], 1 < p < ∞, show that every unit vector in B is an extreme point of B. If X = L∞[a, b], show that the extreme points of B are those functions f ∈ B such that |f | = 1 almost everywhere on [a, b].
- [Topology (Munkres, James Raymond) (Z-Library)] ShowthatinthenitecomplementtopologyonR,everysubspaceiscom-pact. IfRhasthetopologyconsistingofallsetsAsuchthatR−AiseithercountableorallofR,is[0,1]acompactsubspace? ShowthataniteunionofcompactsubspacesofXiscompact.
- [Real Analysis (Halsey L. Royden, Patr... (Z-Library)] The Eberlein-ˇSmulian Theorem . Metrizability of Weak Topologies . X is reexive; (ii) B is weakly compact; (iii) B is weakly sequentially compact.
- [Topology (Munkres, James Raymond) (Z-Library)] SupposethatYiscompactandA={Aα}α∈JisacoveringofYbysetsopeninX. Thenthecollection{Aα∩Y|α∈J}isacoveringofYbysetsopeninY;henceanitesubcollection{Aα1∩Y,. Aαn}isasubcollectionofAthatcoversY.
Cognitive Tools Reference
See .claude/skills/math-mode/SKILL.md for full tool documentation.
版本历史
- d07ff4b 当前 2026-08-20 12:49
