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 n to 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.
1---2name: compactness3description: Problem-solving strategies for compactness in topology4---5
6# Compactness
7
8## When to Use
9
10Use this skill when working on compactness problems in topology.
11
12## Decision Tree
13
14
151. **Is X compact?**
16 - If X subset R^n: Is X closed AND bounded? (Heine-Borel)
17 - If X is metric: Does every sequence have convergent subsequence?
18 - General: Does every open cover have finite subcover?
19 - `z3_solve.py prove "bounded_and_closed"`
20
212. **Compactness Tests**
22 - Heine-Borel (R^n): closed + bounded = compact
23 - Sequential: every sequence has convergent subsequence
24 - `sympy_compute.py limit "a_n" --var n` to check convergence
25
263. **Product Spaces**
27 - Tychonoff: product of compact spaces is compact
28 - Finite products preserve compactness directly
29
304. **Consequences of Compactness**
31 - Continuous image of compact is compact
32 - Continuous real function on compact attains max/min
33 - `sympy_compute.py maximum "f(x)" --var x --domain "[a,b]"`
34
35
36## Tool Commands
37
38### Z3_Bounded_Closed
39```bash
40uv run python -m runtime.harness scripts/z3_solve.py prove "bounded_and_closed"
41```
42
43### Sympy_Limit
44```bash
45uv run python -m runtime.harness scripts/sympy_compute.py limit "a_n" --var n --at oo
46```
47
48### Sympy_Maximum
49```bash
50uv run python -m runtime.harness scripts/sympy_compute.py maximum "f(x)" --var x --domain "[a,b]"
51```
52
53## Key Techniques
54
55*From indexed textbooks:*
56
57- [Topology (Munkres, James Raymond) (Z-Library)] CompactSpaces163 164ConnectednessandCompactnessCh. Itisnotasnaturalorintuitiveastheformer;somefamiliaritywithitisneededbeforeitsusefulnessbecomesapparent. AcollectionAofsubsetsofaspaceXissaidtocoverX,ortobeacoveringofX,iftheunionoftheelementsofAisequaltoX.
58- [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].
59- [Topology (Munkres, James Raymond) (Z-Library)] ShowthatinthenitecomplementtopologyonR,everysubspaceiscom-pact. IfRhasthetopologyconsistingofallsetsAsuchthatR−AiseithercountableorallofR,is[0,1]acompactsubspace? ShowthataniteunionofcompactsubspacesofXiscompact.
60- [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.
61- [Topology (Munkres, James Raymond) (Z-Library)] SupposethatYiscompactandA={Aα}α∈JisacoveringofYbysetsopeninX. Thenthecollection{Aα∩Y|α∈J}isacoveringofYbysetsopeninY;henceanitesubcollection{Aα1∩Y,. Aαn}isasubcollectionofAthatcoversY.
62
63## Cognitive Tools Reference
64
65See `.claude/skills/math-mode/SKILL.md` for full tool documentation.
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