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 the majorant of the set A if a ≥ x for any x ∈ A
a will be a minorant of 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 Rnmultidimensional space
Let the point x0 ∈ Rn and r> 0.
Let's denote an open ball with the center at the point x0 and radius r:
Br(x0) := {x ∈ Rn: ||x-x0|| < r}
Let's choose as a subset of Rn the set A: A ⊂ Rn. Point x0, which belongs to Rn can be characterized as:
The point x0 is the inner point of the set, if such a value can be chosen r0 that there is an open sphere Br(x0) ⊂ A, when r0 > 0
A limit point is a point x0 such that for any r0 the condition Br(x0) ∩ A ≠ ∅ (Br is an open sphere with the center at the point x0 and any radius, the intersection of Br with the set A is not is an empty set)
Boundary point of the set: for any r> 0, the conditions are met Br(x0) ∩ A≠ ∅ and Br(x0) ∩ (Rn\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 x0, there exists a value r0 > 0 such that Br(x0) ⊂ A.
Examples
- An open interval is an open set R
- An open sphere is an open set Rn
- The empty set ∅ and the entire space Rn 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⊂ Rn is open only if A = A°, a special case - the set of interior points is always an open set
- For a set A⊂ Rn, the set of interior points is the intersection of all open sets contained in A
Closed set
A set A⊂ Rn is called closed if the complement Rn\A is an open set
Examples
- Closed intervals are closed sets
- The complement of an open sphere is a closed set ⊂ Rn
- The empty set ∅ and the entire space Rn 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⊂Rn 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⊂ Rn is bounded if there exist r>0 and a point x0 such that A ⊂ Br(x0). 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.
