# 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 15 subcategories, out of 15 total.

### C

- Clopen Sets (15 P)
- Closed Ball is Closed (3 P)
- Closed Mappings (6 P)
- Closed Sets (Metric Spaces) (3 P)

### E

- Empty Set is Closed (4 P)

### I

### R

- Regular Closed Sets (4 P)

### S

### W

- Weakly Hereditary Properties (1 P)

## Pages in category "Closed Sets"

The following 79 pages are in this category, out of 79 total.

### B

### C

- Characterization of Closed Set by Open Cover
- 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 Coarser Topology is Closed in Finer Topology
- 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 Sets of Right Order Space on Real Numbers
- Closed Subspace of Compact Space is Compact
- Closed Subspace of Lindelöf Space is Lindelöf Space
- 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

### F

### 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 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
- Intersection of Regular Closed Sets is not necessarily Regular Closed

### M

### P

- Pasting Lemma for Continuous Mappings on Closed Sets
- Pasting Lemma for Pair of Continuous Mappings on Closed Sets
- Pasting Lemma/Continuous Mappings on Closed Sets
- Pasting Lemma/Counterexample of Infinite Union of Closed Sets
- Pasting Lemma/Finite Union of Closed Sets
- Point at Distance Zero from Closed Set is Element
- Product of Closed Sets is Closed

### 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
- Space of Almost-Zero Sequences is not Closed in 2-Sequence Space
- 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