# Category:Real Number Space

Jump to navigation
Jump to search

This category contains results about the Real Number Space in the context of Metric Spaces.

Let $\R$ be the set of real numbers.

Let $d: \R \times \R \to \R$ be the Euclidean metric on $\R$.

Let $\tau_d$ be the topology on $\R$ induced by $d$.

Then $\left({\R, \tau_d}\right)$ is the **real number space**.

## Subcategories

This category has the following 3 subcategories, out of 3 total.

### R

### U

## Pages in category "Real Number Space"

The following 52 pages are in this category, out of 52 total.

### C

- Closed Intervals form Neighborhood Basis in Real Number Line
- Closed Real Interval is Closed in Real Number Line
- Closed Real Interval is Regular Closed
- Closed Subset of Real Number Space is G-Delta
- Closure of Intersection of Rationals and Irrationals is Empty Set
- Closure of Irrational Numbers is Real Numbers
- Closure of Open Real Interval is Closed Real Interval
- Closure of Rational Numbers is Real Numbers
- Closure of Union of Singleton Rationals is Real Space
- Countable Basis of Real Number Space

### I

### O

- Open Ball in Real Number Line is Open Interval
- Open Rational-Number Balls form Neighborhood Basis in Real Number Line
- Open Real Interval is not Closed Set
- Open Real Interval is not Closed Set/Corollary
- Open Real Interval is Open Ball
- Open Real Interval is Regular Open
- Open Reciprocal-N Balls form Neighborhood Basis in Real Number Line
- Open Sets in Real Number Line

### Q

### R

- Rational Numbers form F-Sigma Set in Reals
- Rational Numbers form Metric Subspace of Real Numbers under Euclidean Metric
- Rationals are Everywhere Dense in Reals
- Real Number is Closed in Real Number Space
- Real Number Line is Complete Metric Space
- Real Number Line is Metric Space
- Real Number Space is First-Countable
- Real Number Space is Lindelöf
- Real Number Space is Locally Compact Hausdorff Space
- Real Number Space is Non-Meager
- Real Number Space is not Countably Compact
- Real Number Space is Paracompact
- Real Number Space is Quasimetric Space
- Real Number Space is Second-Countable
- Real Number Space is Separable
- Real Number Space is Sigma-Compact
- Real Number Space satisfies all Separation Axioms