# General topology

## Topology R *one coordinate*

To begin with, let's consider the topology of space in one coordinate, then expand the knowledge for multidimensional space.

### Intervals

An interval in a single-plane coordinate system, that is, in one in which only the x coordinate is present, is a section of a straight line enclosed between points a and b. If the interval includes points a and b, then the interval it is called closed, if points a and b are not included, then such a section is called an open interval.

Open interval

(a,b) := {x ∈ R : a < x < b}

Closed interval

[a,b] := {x ∈ R : a ≤ x ≤ b}

Also, the interval can be limited only on one side:

(a, ∞) := {x ∈ R : a < x}

(-∞, a) := {x ∈ R : x < a}

### Points

#### Majoranta and minoranta

a will be themajorantof the set A if a ≥ x for any x ∈ A

a will be aminorantof the set A if a≤ x for any x ∈ A

A set is called closed if the set has a majorant and a minor.

#### Maximum and minimum

The maximum of A set A is a point that satisfies the condition {Max(A)=a} = {a > x, x ∈ A}

Minmum of set A: {Min(A) = a} = {a ≤ x, x ∈ A}

#### Supremum and infimum

The supremum is the smallest of all majorants, the infimum is the largest of all minorants.

### Example

A = {x∈R: |x| < 3}, converting, we get {x∈R: -3 < x < 3}. The majorant of the set A can be any point satisfying the condition {x≥3}, for example, 3, 7, etc. of the Minor sets A fits the rule {x≤-3}, for example, -3, -5, etc. The set A has no maximum and there is no minimum. The supremum and infimum of the set A: sup(A) = 3, inf(A) = -3.

## Topology R^{n}*multidimensional space*

Let the point x_{0}∈ R^{n}and r> 0.

Let's denote an open ball with the center at the point x_{0}and radius r:

B_{r}(x_{0}) := {x ∈ R^{n}: ||x-x_{0}|| < r}

Let's choose as a subset of R^{n} the set A: A ⊂ R^{n}. Point
x_{0}, which belongs to R^{n} can be characterized as:

The point x_{0} is the **inner point of the set**, if such a value can be chosen
r_{0} that there is an open sphere B_{r}(x_{0}) ⊂ A,
when r_{0} > 0

**A limit point** is a point x_{0} such that for any r_{0}
the condition B_{r}(x_{0}) ∩ A ≠ ∅ (B_{r} is an open sphere
with the center at the point x_{0} and any radius, the intersection of B_{r} with the set A is not
is an empty set)

**Boundary point of the set**: for any r> 0, the conditions are met
B_{r}(x_{0}) ∩ A≠ ∅ and
B_{r}(x_{0}) ∩ (R^{n}\A) ≠ ∅

The set of interior points of the set A is denoted by A° and is called the interior of the set.

The set of limit points of set A is denoted by A.

The set of boundary points of set A is called the boundary of the set and is denoted by ∂A.

### Open set

A set A is an open set if all its points are internal, that is
for any x_{0}, there exists a value r_{0} > 0 such that B_{r}(x_{0})
⊂ A.

#### Examples

- An open interval is an open set R
- An open sphere is an open set R
^{n} - The empty set ∅ and the entire space R
^{n}are open sets

#### Properties

- The union of an unlimited number of open sets is an open set
- The intersection of open sets is an open set
- A set A⊂ R
^{n}is open only if A = A°, a special case - the set of interior points is always an open set - For a set A⊂ R
^{n}, the set of interior points is the intersection of all open sets contained in A

### Closed set

A set A⊂ R^{n} is called closed if the complement R^{n}\A is an open set

#### Examples

- Closed intervals are closed sets
- The complement of an open sphere is a closed set ⊂ R
^{n} - The empty set ∅ and the entire space R
^{n}are a closed set (and open at the same time)

#### Properties

- The union of an unlimited number of closed sets is a closed set
- The intersection of closed sets is a closed set
- A set A⊂R
^{n}is closed if and only if A = A, a special case is a set limit points are always a closed set

### Bounded set

A set A⊂ R^{n} is bounded if there exist r>0 and a point x_{0} such that
A ⊂ B_{r}(x_{0}). For example, an open sphere is a bounded set, a plane is not
a limited set.

### Compact set

A set is called compact if it is closed and bounded. For example, a closed interval is a compact set.