# Category:Closed Sets

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This category contains results about Closed Sets in the context of Topology.

Definitions specific to this category can be found in Definitions/Closed Sets.

**$H$ is closed (in $T$)** if and only if its complement $S \setminus H$ is open in $T$.

That is, $H$ is **closed** if and only if $\paren {S \setminus H} \in \tau$.

That is, if and only if $S \setminus H$ is an element of the topology of $T$.

This also includes metric spaces.

## Subcategories

This category has the following 9 subcategories, out of 9 total.

### C

### E

### R

### S

### W

## Pages in category "Closed Sets"

The following 77 pages are in this category, out of 77 total.

### B

### C

- Closed Ball is Closed
- Closed Real Interval is Closed in Real Number Line
- Closed Real Interval is Closed Set
- Closed Real Interval is Regular Closed
- Closed Set iff Lower and Closed under Directed Suprema in Scott Topological Ordered Set
- Closed Set in Particular Point Space has no Limit Points
- Closed Set in Topological Subspace
- Closed Set in Topological Subspace/Corollary
- Closed Set is F-Sigma Set
- Closed Set Measurable in Borel Sigma-Algebra
- Closed Sets of Fortissimo Space
- Closed Subspace of Compact Space is Compact
- Closed Unit Interval is not Countably Infinite Union of Disjoint Closed Sets
- Compact Subset of Compact Space is not necessarily Closed
- Compact Subspace of Hausdorff Space is Closed
- Continuity Defined from Closed Sets
- Continuity of Mapping between Metric Spaces by Closed Sets
- Continuous Mapping on Finite Union of Closed Sets

### E

- Empty Set is Closed
- Empty Set is Closed in Metric Space
- Empty Set is Closed in Normed Vector Space
- Empty Set is Closed in Topological Space
- Empty Set is Open and Closed in Metric Space
- Equivalence of Definitions of Closed Set
- Equivalence of Definitions of Closed Set in Metric Space
- Equivalence of Definitions of Closed Set in Normed Vector Space

### F

- F-Sigma Set is not necessarily Closed Set
- Finite Subset of Metric Space is Closed
- Finite Subset of Normed Vector Space is Closed
- Finite Subset of Space is Closed
- Finite Subspace of Hausdorff Space is Closed
- Finite Union of Closed Sets is Closed
- Finite Union of Closed Sets is Closed/Normed Vector Space
- Finite Union of Closed Sets is Closed/Topology

### H

### I

- Infinite Union of Closed Sets of Metric Space may not be Closed
- Interior of Closed Real Interval is Open Real Interval
- Interior of Closed Set of Particular Point Space
- Intersection of Closed Set with Compact Subspace is Compact
- Intersection of Closed Sets is Closed
- Intersection of Closed Sets is Closed/Normed Vector Space
- Intersection of Closed Sets is Closed/Topology
- Intersection of Regular Closed Sets is not necessarily Regular Closed

### M

### N

### S

- Set Closure is Smallest Closed Set/Normed Vector Space
- Set is Closed iff Equals Topological Closure
- Set is Closed iff it Contains its Boundary
- Set is Closed in Metric Space iff Closed in Induced Topological Space
- Space is Closed in Itself
- Subset of Euclidean Plane whose Product of Coordinates are Greater Than or Equal to 1 is Closed
- Subset of Metric Space contains Limits of Sequences iff Closed
- Subset of Metric Space is Closed iff contains all Zero Distance Points
- Subset of Particular Point Space is either Open or Closed