compactness

Compactness

When to Use

Use this skill when working on compactness problems in topology.

Decision Tree

1. Is X compact?
2. If X subset R^n: Is X closed AND bounded? (Heine-Borel)
3. If X is metric: Does every sequence have convergent subsequence?
4. General: Does every open cover have finite subcover?
5. z3_solve.py prove "bounded_and_closed"

1. Compactness Tests
2. Heine-Borel (R^n): closed + bounded = compact
3. Sequential: every sequence has convergent subsequence
4. sympy_compute.py limit "a_n" --var n to check convergence

1. Product Spaces
2. Tychonoff: product of compact spaces is compact
3. Finite products preserve compactness directly

1. Consequences of Compactness
2. Continuous image of compact is compact
3. Continuous real function on compact attains max/min
4. 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.