Real line

The order the natural numbers shown on the numerical straight.

The real line or number line is a one-dimensional geometric construction, or straight line, which contains all real numbers either through a one-to-one correspondence or through a Bijective application, used to represent numbers as specially marked points, for example integers by means of a line called line graduated as the integer of ordered and separated with the same distance.

The real line is naturally divided into two identical and symmetrical halves with respect to the origin, that is, the number zero. In addition, this number line is a line on which integers are usually graphed as points that are a uniform distance apart. It allows us to locate and compare numbers as well as perform addition and subtraction operations.

Topologies on the real line

In order to understand the internal structure of said line, different topologies can be defined under which the real line has topological and geometric properties, different from the usual metric topology.

Usual topology

The number line is considered to be made up of points and intervals.

Interior point

Be H a subset of R. A point and0{displaystyle and} of H is called a inner point of H, if there is a positive real r such that₀ - r, y_or +r 한. A. The set of H's inner points is named interior de H, denotes by int(a). If the point and₀ It's inside A, it'll be said that A is environment from that point.

Example: Yes H = {1}[[3,5].[6, 8/2005. Points 1, 3, 5 and 6 are not internal points of H. While int(H) = ₡3,5 backwards 6, 8 PHP.

Note that if H is part of J then the interior of H is part of the interior of J. Also that the interior of H is part of H.

Open complex

A subset K of ℝ is called open, if every point of K is an interior point of K. That is, K ⊂ Int(K).

It is obvious that R and ∅ are open set.

Any open interval ≤m, n/2003/RR is an open subset of R

The intersection of θ-1, 1/n/2005 with θ-1/n, 1 plan is an open subset of R, for any positive integer

≤2, 8 plan - [4, 6] is an open subset of R.

For any set of real numbers your interior is an open set.

Topological properties

1. The union of an open family of R is an open one.
2. The intersection of two R opens is an open R(considering the empty as open set).
3. The arbitrary intersection of open infinity does not have to be an open one.
4. The intervals ≤m, -≤3⁄2⁄4⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2⁄2₊.