# Category:Compact Spaces

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This category contains results about **Compact Spaces** in the context of **Topology**.

Definitions specific to this category can be found in Definitions/Compact Spaces.

A topological space $T = \struct {S, \tau}$ is **compact** if and only if every open cover for $S$ has a finite subcover.

## Subcategories

This category has the following 52 subcategories, out of 52 total.

### C

- Compactifications (empty)
- Countably Metacompact Spaces (6 P)
- Countably Paracompact Spaces (5 P)

### E

### F

- Fully Normal Spaces (4 P)
- Fully T4 Spaces (6 P)

### H

- Heine-Borel Theorem (10 P)
- Heine-Cantor Theorem (3 P)
- Hereditarily Compact Spaces (3 P)
- Hilbert Cube is Compact (2 P)

### L

### M

### N

### O

### P

- Proper Mappings (empty)
- Pseudocompact Spaces (11 P)

### R

- Relatively Compact Subspaces (4 P)

### S

- Sierpiński's Theorem (2 P)
- Sigma-Locally Compact Spaces (empty)
- Stone-Weierstrass Theorem (3 P)

### T

- Tychonoff's Theorem (8 P)

### W

- Weakly Countably Compact Spaces (14 P)

## Pages in category "Compact Spaces"

The following 154 pages are in this category, out of 154 total.

### C

- Cantor Space is Compact
- Closed and Bounded Subset of Normed Vector Space is not necessarily Compact
- Closed and Bounded Subspace is not necessarily Compact
- Closed Ordinal Space is Compact
- Closed Real Interval is Compact
- Closed Subspace of Compact Space is Compact
- Closed Subspace of Lindelöf Space is Lindelöf Space
- Closure in Infinite Particular Point Space is not Compact
- Coarser Topology than Compact Space is Compact
- Compact Complement Topology is Coarser than Euclidean Topology
- Compact Complement Topology is Compact
- Compact First-Countable Space is Sequentially Compact
- Compact Hausdorff Space is Locally Compact
- Compact Hausdorff Space is T4
- Compact Hausdorff Space with no Isolated Points is Uncountable/Lemma
- Compact Hausdorff Topology is Maximally Compact
- Compact Hausdorff Topology is Minimal Hausdorff
- Compact in Subspace is Compact in Topological Space
- Compact Metric Space is Complete
- Compact Metric Space is Separable
- Compact Metric Space is Totally Bounded
- Compact Set of Irrational Numbers is Nowhere Dense
- Compact Set of Rational Numbers is Nowhere Dense
- Compact Sets in Countable Complement Space
- Compact Sets in Fortissimo Space
- Compact Space in Particular Point Space
- Compact Space is Countably Compact
- Compact Space is Lindelöf
- Compact Space is Paracompact
- Compact Space is Pseudocompact
- Compact Space is Sigma-Compact
- Compact Space is Strongly Locally Compact
- Compact Space is Weakly Locally Compact
- Compact Space is Weakly Sigma-Locally Compact
- Compact Space satisfies Finite Intersection Axiom
- Compact Subset of Compact Space is not necessarily Closed
- Compact Subsets of T3 Spaces
- Compact Subspace of Hausdorff Space is Closed
- Compact Subspace of Linearly Ordered Space
- Compact Subspace of Metric Space is Bounded
- Compact Subspace of Metric Space is Sequentially Compact in Itself
- Compact Subspace of Real Numbers is Closed and Bounded
- Compact Subspace of Topological Vector Space is von Neumann-Bounded
- Compactness and Sequential Compactness are Equivalent in Metric Spaces
- Compactness is Preserved under Continuous Surjection
- Compactness Properties in Hausdorff Spaces
- Compactness Properties in T3 Spaces
- Compactness Properties Preserved under Continuous Surjection
- Compactness Properties Preserved under Projection Mapping
- Cone on Compact Space is Compact
- Continuous Bijection from Compact to Hausdorff is Homeomorphism
- Continuous Bijection from Compact to Hausdorff is Homeomorphism/Corollary
- Continuous Function from Compact Hausdorff Space to Itself Fixes a Non-Empty Set
- Continuous Function on Compact Space is Bounded
- Continuous Image of Compact Space is Compact
- Continuous Mapping from Compact Space to Hausdorff Space is Closed Mapping
- Continuous Mapping from Compact Space to Hausdorff Space Preserves Local Connectedness
- Continuous Mappings preserve Compact Subsets
- Continuous Real-Valued Function on Compact Space is Bounded
- Countably Compact Lindelöf Space is Compact
- Countably Compact Metric Space is Compact

### D

- Dilation of Compact Set in Topological Vector Space is Compact
- Discrete Space is Compact iff Finite
- Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods
- Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods/Lemma
- Distance between Disjoint Compact Set and Closed Set in Metric Space is Positive
- Double Pointed Finite Complement Topology is Compact

### E

- Either-Or Topology is Compact
- Empty Set is Compact Space
- Equivalence of Definitions of Compact Topological Space
- Equivalence of Definitions of Compact Topological Subspace
- Excluded Point Space is Compact
- Existence of Compact Hausdorff Space which is not T5
- Existence of Compact Space which is not Sequentially Compact
- Existence of Compact Space which Satisfies No Separation Axioms
- Existence of Maximal Compact Topological Space which is not Hausdorff
- Existence of Minimal Hausdorff Space which is not Compact
- Existence of Paracompact Space which is not Compact
- Existence of Sigma-Compact Space which is not Compact

### F

### H

- Hahn-Banach Separation Theorem/Normed Vector Space/Real Case/Compact Convex Set and Closed Convex Set
- Heine-Borel iff Dedekind Complete
- Heine-Borel Theorem
- Heine-Borel Theorem/Metric Space
- Heine-Borel Theorem/Normed Vector Space
- Heine-Borel Theorem/Real Line
- Heine-Cantor Theorem
- Hilbert Cube is Compact

### I

- Infinite Particular Point Space is not Compact
- Infinite Set in Compact Space has Omega-Accumulation Point
- Intersection of Closed Set with Compact Subspace is Compact
- Intersection of Compact and Closed Subsets of Normed Finite-Dimensional Real Vector Space with Euclidean Norm is Compact
- Intersection of Nested Closed Subsets of Compact Space is Non-Empty

### M

### N

### O

### Q

### R

### S

- Second-Countable Space is Compact iff Countably Compact
- 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
- Sequentially Compact Metric Space is Compact
- Set of 2-Dimensional Indefinite Real Orthogonal Matrices is not Compact in Normed Real Square Matrix Vector Space
- Set of 2-Dimensional Real Orthogonal Matrices is Compact in Normed Real Square Matrix Vector Space
- Set of Integers is not Compact
- Set of Inverse Positive Integers with Zero is Compact
- Shift of Finite Type is Compact
- Sierpiński's Theorem
- Singleton Set in Discrete Space is Compact
- Space is Compact iff exists Basis such that Every Cover has Finite Subcover
- Spectrum of Bounded Linear Operator is Compact
- Stone-Weierstrass Theorem
- Subset of Indiscrete Space is Compact
- Subset of Indiscrete Space is Compact and Sequentially Compact
- Subspace of Finite Complement Topology is Compact
- Sum of Compact Subsets of Topological Vector Space is Compact

### U

- Unbounded Set of Real Numbers is not Compact
- Unbounded Set of Real Numbers is not Compact/Normed Vector Space
- Uniform Continuity on Metric Space does not imply Compactness
- Union of Image of Convergent Sequence and Limit in Topological Space is Compact
- Union of Two Compact Sets is Compact
- Upper Closure is Compact in Topological Lattice