Category:Compact Spaces

C

• [+] Continua‎ (1 C)
• [×] Countably Compact Spaces‎ (24 P)
• [×] Countably Metacompact Spaces‎ (6 P)
• [×] Countably Paracompact Spaces‎ (4 P)

F

• [×] Fully Normal Spaces‎ (4 P)
• [×] Fully T4 Spaces‎ (4 P)

L

• [×] Lindelöf Spaces‎ (21 P)
• [+] Locally Compact Spaces‎ (1 C, 11 P)

M

• [×] Metacompact Spaces‎ (5 P)

P

• [×] Paracompact Spaces‎ (15 P)
• [×] Pseudocompact Spaces‎ (7 P)

S

• [×] Sequentially Compact Spaces‎ (19 P)

S cont.

• [+] Sigma-Compact Spaces‎ (1 C, 12 P)
• [×] Sigma-Locally Compact Spaces‎ (2 P)
• [×] Strongly Locally Compact Spaces‎ (6 P)

W

• [×] Weakly Countably Compact Spaces‎ (13 P)

A

• Alexander's Compactness Theorem
• Arens-Fort Space is Not Compact

C

• Closed Subspace of Compact Space is Compact
• Closure in Infinite Particular Point Space is Not Compact
• Compact Complement Topology is Coarser than Euclidean Topology
• Compact Complement Topology is Compact
• Compact Hausdorff Space is T4
• Compact Hausdorff Topology is Maximally Compact
• Compact Hausdorff Topology is Minimal Hausdorff
• Compact Metric Space is Complete
• Compact Metric Space is Totally Bounded
• Compact Sets in Countable Complement Space
• Compact Space in Particular Point Space
• Compact Space is Countably Compact
• Compact Space is Locally Compact
• Compact Space is Paracompact
• Compact Space is Sigma-Compact
• Compact Space is Sigma-Locally Compact
• Compact Space is Strongly Locally Compact
• Compact Space satisfies Finite Intersection Axiom
• Compact Subsets of T3 Spaces

C cont.

• Compact Subspace of Hausdorff Space is Closed
• Compact Subspace of Metric Space is Bounded
• Compactness Preserved under Continuous Surjection
• Compactness Properties in Hausdorff Spaces
• Compactness Properties in T3 Spaces
• Compactness Properties Preserved under Continuous Surjections
• Compactness Properties Preserved under Projection Mapping
• Continuous Bijection from Compact to Hausdorff is Homeomorphism
• Continuous Image of a Compact Space is Compact
• Countably Compact Lindelöf Space is Compact
• Countably Compact Metric Space is Compact
• Countably Compact Metric Space is Compact/Proof 1
• Countably Compact Metric Space is Compact/Proof 2

D

• Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods
• Double Pointed Finite Complement Topology is Compact

E

• Equivalent Definitions of Compactness
• Excluded Point Space is Compact

F

• Finite Discrete Space is Compact
• Finite Discrete Space Satisfies All Compactness Properties
• Finite Space Satisfies All Compactness Properties
• Finite Topological Space is Compact

F cont.

• Fort Space is Compact
• Fortissimo Space is Not Compact

I

• Infinite Particular Point Space is Not Compact

M

• Metacompact Countably Compact Space is Compact
• Metrizable iff Regular and has Sigma-Locally Finite Basis
• Modified Fort Space is Compact

N

• Neighborhood of Origin in Arens-Fort Space is Not Compact

O

• Open Extension Space is Compact

S

• Second-Countable Space: Compactness Equivalent to Countable Compactness
• Sequence of Implications of Global Compactness Properties
• Sequence of Implications of Local Compactness Properties
• Sequence of Implications of Metric Space Compactness Properties
• Sequence of Implications of Paracompactness Properties
• Sequences in a Compact Space Have a Convergent Subsequence
• Sequentially Compact Metric Space is Compact
• Sequentially Compact Metric Space is Compact/Proof 1
• Sequentially Compact Metric Space is Compact/Proof 2
• Singleton Sets in Discrete Space are Compact
• Subspace of Finite Complement Topology is Compact

T

• Tychonoff's Theorem