# 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 $T = \left({S, \tau}\right)$ is **compact** if and only if every open cover for $S$ has a finite subcover.

## Subcategories

This category has the following 35 subcategories, out of 35 total.

### C

### E

### F

### H

### L

### M

### P

### S

### U

### W

## Pages in category "Compact Spaces"

The following 121 pages are in this category, out of 121 total.

### C

- Cantor Space is Compact
- Closed Ordinal Space is Compact
- Closed Real Interval is Compact
- Closed Subspace of Compact Space is Compact
- Closure in Infinite Particular Point Space is not Compact
- Definition:Compact (Euclidean Space)
- 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 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 Space in Particular Point Space
- Compact Space is Countably Compact
- Compact Space is Paracompact
- 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
- Compactness from Basis
- 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
- Continuous Bijection from Compact to Hausdorff is Homeomorphism
- Continuous Bijection from Compact to Hausdorff is Homeomorphism/Corollary
- Continuous Function on Compact Space is Bounded
- Continuous Image of Compact Space is Compact
- Continuous Image of Compact Space is Compact/Corollary 1
- Continuous Image of Compact Space is Compact/Corollary 2
- Continuous Image of Compact Space is Compact/Corollary 3
- Continuous Image of Compact Space is Compact/Corollary 3/Proof 1
- Continuous Image of Compact Space is Compact/Corollary 3/Proof 2
- Continuous Mapping from Compact Space to Hausdorff Space Preserves Local Connectedness
- Countably Compact Lindelöf Space is Compact
- Countably Compact Metric Space is Compact

### D

### E

- 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 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

### I

### M

### N

### O

### Q

### 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 Integers is not Compact
- Sierpiński's Theorem
- Singleton Set in Discrete Space is Compact
- Subset of Indiscrete Space is Compact
- Subset of Indiscrete Space is Compact and Sequentially Compact
- Subspace of Finite Complement Topology is Compact