Compactness

When to Use

Decision Tree

1. 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"
2. 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
3. Product Spaces
   • Tychonoff: product of compact spaces is compact
   • Finite products preserve compactness directly
4. 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

• [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