Lecture
Space
Metric space.
Let there be some set of elements Ω. To each pair (x,y) let us assign 
and call it the distance between x and y.
and 
, if x=y
= 
≤
+ 
x,y
=│x-y│
x,y 
x=(
,
,
,…,
) y=(
,
,…
)
- Euclidean metric
For k
= 
|=
Any metric space can be denoted (Ω,
).
An open ball with center at point a and radius r: 
<r
≤r – closed ball.
– sphere of radius with a neighbourhood at point a.
A neighbourhood of point a is called a set in which a ball can be placed. It contains the subset: 
c
One can introduce the concept of a limit. Let a sequence 
be given, and we will say that 
=a 
as k
Topological space (“topos” place).
Let Ω - be a set of elements, and let us call a topology a system of subsets(∑) that satisfies the following conditions:
,
,
If 
- is a subset
, then 
is called a finer topology, and 
a coarser one. Sets belonging to 
are called open.
A neighbourhood of the point x will be any set such that 
and x
.
x
U(x) any and x
Example 1: Let a set Ω be given. Consider the topology ∑={Ω,
}.
The neighbourhood is the entire set Ω. Example 1 is the coarsest topology.
The finest topology is the space of discrete points.∑ - the set of all subsets of Ω.
Example 2: Ω={a,b}
∑= {, {a, b}, {b}-open set}
{a} – closed set
If a metric space is given, a topology can be constructed from it.
∑={0,Ω,
Linear space.
Let some set E be called a linear (or vector) space if it satisfies the following axioms:
z=x+y
Commutativity: x+y=y+x
Associativity: (x+y)+z=x+(y+z)
: x+0=x
”: x+(-x)=0
Normed spaces.
It is obtained from a linear space by introducing a norm.
A space 
is called a normed space if
a 
-norm is given 
, which satisfies the following axioms:
1) 
and 
(non-degeneracy)
2) 
(homogeneity)
3) 
(Triangle inequality)
A norm can be interpreted as the length of an element.
Example: Let us introduce 
Sequence space
: 
: 
: 
– the space of functions continuous on 
.
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