**Usual topology of R** refers to one of the many topologies that it is possible to construct in the set **R**of the real numbers. To calculate the limit of a real function at a point this point is required to be an accumulation point, if we want to study the continuity of a function over an interval, a closed interval is preferred, which tells us that such a function is also continuous in the ends of the range. Situations of limit, continuity, allow us to generalize these concepts on parts of the line (real numbers) and they will acquire more abstract and general roles. Therefore we are going to build a topology of R (the set of real numbers), for which using open intervals and the union of them, we need to define new sets, which must form a new mathematical structure of R.

## Summary

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- 1 Open
- 1 Usual topology

- 2 Closed
- 3 Environment
- 4 Adhesion, interior, accumulation point of a part of R
- 1 Example
- 2 Characterization of Q and its complement

- 5 Exterior and border of a part of R
- 6 Applications
- 1 Proposition
- 2 Definition

- 7 See also
- 8 References
- 9 Sources
- 10 External links

## Open

Call **open** to any union of open intervals; these satisfy:

- Any open junction is an open,
- any finite open intersection is an open,
- ∅ and R are open

### Usual topology

The set T _{u} of all openings of R is called the **usual Topology** of R and the pair (R, T _{u} ) is named **Topological Space** of Real Numbers.

## Closed

**Any** complement to an open is called **closed** ; closed ones satisfy

- Any closed intersection is a closed,
- any finite closed union is a closed,
- ∅ and R are closed, this flows according to Augustus de Morgan’s laws.

## Environment

It is called **environment** of a, real number, any part that contains an open that, in turn, contains the real *a* .

Denoted by N (a) is the set of all environments from a.

An open is the environment of all its points.

## Adhesion, interior, point of accumulation of a part of R

Is said *to* , element R, is **point bonding** of R Besides, if

for every environment V of a, the intersection of V and A is non-empty.

the **adhesion** A ^{–} is the set of all the adherent points of A. It can still be said that it is the smallest closure that contains A.

h is said to be the **interior point** of A if there is an open that contains *h* and is contained in A. The **interior** A ^{º} is the set of all interior points of A. The interior of A is the largest open set contained in TO.

a, element of R, is **accumulation point** of A, if

for every environment V of a, the intersection of A and V is non-empty and said intersection is different from {a}

k, element of R, is **an isolated point** of A if it is in A and there is no environment that contains it and is contained in A.

### Example

let the set M = [3,8) U {1}

adherence of M is [3.8]

interior of M is (3.8)

8 is an accumulation point of M, as is 5 or 3

1 is an isolated point of M.

### Characterization of Q and its complement

The set of all rational numbers is designated by Q and the irrational ones, complement of Q, by Q ^{c} , for which we have:

Both Q y and R \ Q = Q ^{c} have their adherence equal to R. The intersection of an environment of a rational number with R is non-empty.

Likewise the set Q and its complement the set of the irrationals have their interior = ∅

Any rational or irrational number is the point of accumulation of any open interval that contains it.

## Exterior and border of a part of R

We will say that point *e* is the outer point of A, a subset of R, if *e* is the inner point of the complement; that is, there is an environment of *e* , not that it is not contained in A.

the set of the exterior points of A, is called the **exterior** of A.

We say that *f* is a **border point** A, if any environment *f* contains points inside and outside the set A. It is called **border** A to all border points A.

## Applications

As an example of the use of a topological concept, we will present a

### Proposition

The sequence (x _{n} ) converges towards L if, only if, for each environment V of L there exists a positive integer m (V) such that x _{n} is in V whatever is n ≥ m (V) ^{[1]} .

As another application, the concept of continuity is proposed:

### Definition

F is a function defined on a subset S of **R** and values in **R** . F is said to be **continuous** at point *a* if, for any environment W of f (x), the set f ^{-1} (W) is an environment of *a* relative to S.