# Category:Real Number Line with Euclidean Topology

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This category contains results about **Real Number Line with Euclidean Topology**.

Definitions specific to this category can be found in Definitions/Real Number Line with Euclidean Topology.

Let $\R$ denote the real number line.

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

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

The topology $\tau_d$ induced by $d$ is called the **Euclidean topology**.

Hence $\struct {\R, \tau_d}$ is referred to as the **real number line with the Euclidean topology**.

## Subcategories

This category has the following 5 subcategories, out of 5 total.

## Pages in category "Real Number Line with Euclidean Topology"

The following 37 pages are in this category, out of 37 total.

### C

- Closed Real Interval is Closed in Real Number Line
- Closed Real Interval is Regular Closed
- Closed Subset of Real Number Line 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 Number Line
- Countable Basis of Real Number Line

### I

### Q

### R

- Rational Numbers form F-Sigma Set in Reals
- Rationals are Everywhere Dense in Reals/Topology
- Real Number is Closed in Real Number Line
- Real Number Line is First-Countable
- Real Number Line is Lindelöf
- Real Number Line is Locally Compact Hausdorff Space
- Real Number Line is Non-Meager
- Real Number Line is not Countably Compact
- Real Number Line is Paracompact
- Real Number Line is Second-Countable
- Real Number Line is Separable
- Real Number Line is Sigma-Compact
- Real Number Line satisfies all Separation Axioms
- Real Number Line with Off-Center Distance Function is Quasimetric Space