# Category:Set Closures

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This category contains results about Set Closures in the context of Topology.

Definitions specific to this category can be found in Definitions/Set Closures.

The **closure of $H$ (in $T$)** is defined as:

- $H^- := H \cup H'$

where $H'$ is the derived set of $H$.

## Subcategories

This category has the following 15 subcategories, out of 15 total.

### C

### E

### K

### S

## Pages in category "Set Closures"

The following 89 pages are in this category, out of 89 total.

### C

- Characterization of Closure by Basis
- Characterization of Closure by Open Sets
- Closed Image of Closure of Set under Continuous Mapping equals Closure of Image
- Closure Equals Union with Derivative
- Closure of Cartesian Product is Product of Closures
- Closure of Connected Set is Connected
- Closure of Convex Subset in Normed Vector Space is Convex
- Closure of Dense-in-itself is Dense-in-itself in T1 Space
- Closure of Derivative is Derivative in T1 Space
- Closure of Empty Set is Empty Set
- Closure of Finite Union equals Union of Closures
- Closure of Half-Open Real Interval is Closed Real Interval
- Closure of Image under Continuous Mapping is not necessarily Image of Closure
- Closure of Infinite Subset of Finite Complement Space
- Closure of Infinite Union may not equal Union of Closures
- Closure of Integer Reciprocal Space
- Closure of Interior of Closure of Union of Adjacent Open Intervals
- Closure of Intersection is Subset of Intersection of Closures
- Closure of Intersection may not equal Intersection of Closures
- Closure of Intersection of Rationals and Irrationals is Empty Set
- Closure of Irrational Interval is Closed Real Interval
- Closure of Irrational Numbers is Real Numbers
- Closure of Non-Empty Bounded Subset of Metric Space is Bounded
- Closure of Open Real Interval is Closed Real Interval
- Closure of Open Set of Closed Extension Space
- Closure of Open Set of Particular Point Space
- Closure of Preimage under Continuous Mapping is not necessarily Preimage of Closure
- Closure of Rational Numbers is Real Numbers
- Closure of Real Interval is Closed Real Interval
- Closure of Set of Condensation Points equals Itself
- Closure of Subset in Subspace
- Closure of Subset in Subspace/Corollary 1
- Closure of Subset in Subspace/Corollary 2
- Closure of Subset of Closed Set of Metric Space is Subset
- Closure of Subset of Closed Set of Topological Space is Subset
- Closure of Subset of Double Pointed Topological Space
- Closure of Subset of Indiscrete Space
- Closure of Subset of Metric Space is Closed
- Closure of Subset of Metric Space is Intersection of Closed Supersets
- Closure of Subspace of Normed Vector Space is Subspace
- Closure of Topological Closure equals Closure
- Closure of Union contains Union of Closures
- Closure of Union of Adjacent Open Intervals
- Complement of Closure is Interior of Complement
- Complement of Interior equals Closure of Complement
- Condition for Point being in Closure
- Condition for Point being in Closure/Metric Space
- Continuity Defined by Closure

### D

### I

- Infimum of Bounded Below Set of Reals is in Closure
- Interior equals Complement of Closure of Complement
- Interior is Subset of Interior of Closure
- Interior of Closure of Interior of Union of Adjacent Open Intervals
- Intersection of Open Set with Closure of Set is Subset of Closure of Intersection
- Isolated Point of Closure of Subset is Isolated Point of Subset

### K

- Kuratowski's Closure-Complement Problem
- Kuratowski's Closure-Complement Problem/Closure
- Kuratowski's Closure-Complement Problem/Closure of Interior
- Kuratowski's Closure-Complement Problem/Closure of Interior of Closure
- Kuratowski's Closure-Complement Problem/Closure of Interior of Complement
- Kuratowski's Closure-Complement Problem/Interior of Closure
- Kuratowski's Closure-Complement Problem/Interior of Closure of Interior
- Kuratowski's Closure-Complement Problem/Proof of Maximum

### S

- Set between Connected Set and Closure is Connected
- Set Closure as Intersection of Closed Sets
- Set Closure is Smallest Closed Set
- Set Closure is Smallest Closed Set/Topology
- Set Closure Preserves Set Inclusion
- Set is Closed iff Equals Topological Closure
- Set is Subset of its Topological Closure
- Set together with Condensation Points is not necessarily Closed
- Set together with Omega-Accumulation Points is not necessarily Closed
- Subset of Metric Space is Subset of its Closure
- Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space
- Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space/Necessary Condition
- Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space/Sufficient Condition
- Supremum of Bounded Above Set of Reals is in Closure