We completed construction of the set of real numbers by completing the set of rational numbers 
The set of real numbers 
The Cartesian powers 
Once the distance is defined, we can talk about (round, open) balls of the form 
The rest of the class was devoted to developing and accurate formalization of the intuitive idea of continuity of a map 
Lecture notes (accumulated): https://drive.google.com/file/d/1knEouBLBIGD61dyYLU5NKx8MOHTEhziq/view?usp=sharing
: the notation 
means that a set denoted by 
is an element of another set denoted by 
. We say that the two sets are equal if and only if they consist of the same elements. There is a unique set 
which has no elements at all. 
: we say that a set 
and write 
, if any element 
is also an element of 
. By definition, 
and intersection 
of any two sets. It is important to stress that there is no problem with defining infinite unions and intersections! 
which may or may not include the number 
(this is a convention). The set of Peano axioms which starts, say, with two elements 
and for any element 
introduces a unique element 
, the successor of 
), allows us to define the set of natural numbers 
“inductively” (imitating the human process of counting) and immediately leads us to the conclusion that 
and 
are sometimes solvable with respect to 
in the natural numbers. Yet we can extend the set of naturals 
of integer numbers, closed by subtraction, and the set 
is always solvable if 
(such a set is called a field with the four arithmetic operations) and can be equipped with the complete order 
. 
expressing the length of the diagonal of a unit square. In the same way the circumference 
of the unit circle is also not a rational number. To cope with this, we need to extend the rational numbers even more, this time adding to them solutions of all (infinite) systems of two-sided inequalities of the form 
with 
. On this way we construct the set of real numbers 
and variance 
the normalized sum 
converges to the so called normal distribution with the density function 
.
we introduced two very important numeric characteristics, expectation and variance.
is (in the simplest setting) the weighted sum of values which the variable 
takes with weights equal to the corresponding probabilities; it is a finite number or can be 
. If the expectation is zero, this means that 
describes the spread of random variable around its expected value 
) equal to the expectation of the square 
. Small variance means that 
its preimage 
must be an admissible (random) event. 
. If necessary, we add indication of the random variable and write 
. 
and locally near a point 
of two close points 
in the domain are close to each other in the target space of a function 
. Yet this formulation lacks quantifiers and a careful look at how they can be placed reveals several close but not identical definitions (e.g., continuity at every point of the domain is weaker than the uniform continuity on the domain). 
at a point 
) is nothing more than the possibility of extending the “natural” domain of this function (given, say, by an explicit function) by assigning the value 
so that the extended function retains continuity on the larger set 
. Usually such procedure is used to “avoid” division by zero in the expressions like 
on the diagonal 
or 
at the origin, that will appear when introducing derivatives later. 
or their proper subsets. This is where the topology with its specific language comes to help and one can introduce and study various properties of continuous functions in the general context. Thus, for instance, one can describe “bounded” (compact) sets in absence of any distance function, and prove that this class is closed by actions of continuous functions (maps). In a similar way one can formalize the notion of a connected set (meaning that this set consists of a “single piece”) and show that continuous functions cannot destroy continuity by “tearing apart” such sets. The proofs in this “abstract context” are very simple, often one-liners and may seem to be a shallow playing with abstract words, but it was a cautious crafting of numerous definitions that made the proofs easy, and in practical applications one needs to verify that all functions and sets satisfy exactly the appropriate definition. 
at a point 
: the closer a point 
. Of course, this completely transparent phrase of a human language requires quantifiers to measure “proximity” and “smallness” and how one implies the other. 
is a set 
equipped with two operations, the vector sum/difference 
, a binary commutative invertible operation (making 
. 
between two (in general, different) vector spaces is called linear, if it “respects” both linear operations. In particular, it must map the zero vector of 
. If 
, then all linear functions have a simplest form 
for some number 
(possibly, zero), in the general multidimensional case a linear map is determined by 
real numbers which are naturally arranged in the form of a matrix, an 
-table. 
, is not linear (
unless 
, in which case the shift becomes a dull identical map). Thus we need to make one last step and consider the smallest class of maps that would contain all linear maps and all shifts (parallel translations) and closed by compositions. This class is called the class of affine maps: reducing similar terms, one can show that by definition, an affine map 
is a composition, 
, where 
is a linear map and $T$ a translation in the target space. Explicitly, we have 
(“linear non-homogeneous function”). Note that there is a well-established tradition not to enclose the argument of linear maps in the parentheses and write 
rather than 
: this is because one can treat 
? Let us start from the simplest case 
(functions of one real variable). If 
for some two constants 
, which could be found using the values of 
at any two different points 
by the formulas 
do not depend on the choice of the two points 
. 
there exists the limit 
. Then the function 
, in the following sense: the relative error of approximation 
tends to zero as 
, 
. 
is called the derivative of 
the differential of 
. If 
), while the differential will become a function of two independent arguments 
(usually denoted by 
without extra braces, or simply 
, omitting both arguments. 
a metric on it which we (only for simplicity!) will assume translation invariant and denote 
. Everywhere below 
will denote the open ball of radius 
centered at a point 
: 
. 
such that the open ball 
lies in 
. This is the same as saying that 
.
is called exterior point, if it is interior for the complement 
. The points that are neither interior nor exterior are called the boundary points of 
and 
respectively. 
is called an accumulation point (for 
such that 
. The set of all accumulation points for 
. Sometimes the notation 
is used for the closure. 
. Prove that the exterior of 
.
. The subset is closed, if it coincides with its own closure 
.
and 
. Describe interior, closure and boundary of this segment. Is it open? closed? neither?
and 
are both open and closed simultaneously. 
is an open subset 
.
and its preimage 
. Let 
be any point in this preimage: by definition, this means that 
. Since 
, there exists a ball 
of positive radius 
which lies in 
, that is, all points of the ball 
are mapped inside 
together with the point 
. Consider an arbitrary open ball 
of positive radius 
is called open relative to 
such that 
. 
the value 
defined on a subset 
of any open subset 
. There are only three axioms these open sets must obey: 
between any two topological spaces, with an immediate trivial corollary that composition of any two continuous maps (when defined) will again be continuous. This becomes a trivial observation (why?), although the proof in the “classical” case is also very easy. 
with 
. If such representation is impossible, we call the space connected. Examples are numerous: the Euclidean spaces of all (finite) dimensions are connected, yet the set 
is disconnected, as the two relatively open subsets provide the partition. 
and 
are obviously (relatively) open and disjoint from each other, but their union is the whole of 
. 
are two disjoint open subsets such that 
. Then their preimages 
and 
which are connected topological spaces with respect to the relative topology inherited from 
(finite or infinite, doesn’t matter) takes two different values 
, then it takes also all intermediate values 
. 
by the continuous map 
, 
, is the disconnected union 
. 
is an isolated point of a topological space 
is both open and closed
do not allow such choice: the only nonnegative element that is smaller than all number in the above set, is zero which is non-positive. 
is an open covering of the topological space 
are open, and
.
is a collection of open subsets 
in 
. 
, that is, a collection of open sets which still cover 
, that is, discarding (throwing away) some open sets from the initial covering. 
is a function continuous at all points of its domain. Then for every point 
such that 
. The collection 
is an open covering of 
.
is not covering of ![[n,n+1]](https://s0.wp.com/latex.php?latex=%5Bn%2Cn%2B1%5D&bg=ffffff&fg=000000&s=0&c=20201002)
will become uncovered. Yet some of the coverings are definitely non-minimal, and one can safely remove some of the open sets which were used to cover.
(it is non-compact because the infinite open covering 
cannot be reduced to a finite subcovering) can be compactified by adding two endpoints 
and 
. The explanation “on one leg” of this phenomenon is simple: adding the extra points imposes additional requirement on the collection of open sets to be a covering, i.e., to cover the extra point as well. 
. This is a covering, since each point belongs to “its” ball. Yet no finite subcovering can be selected from this covering: the union of finitely many balls of radius 1 must be bounded. A similar easy argument shows that a set which is not closed, also cannot be compact. ![[0,1]\subseteq\mathbb R^1](https://s0.wp.com/latex.php?latex=%5B0%2C1%5D%5Csubseteq%5Cmathbb+R%5E1&bg=ffffff&fg=000000&s=0&c=20201002)
is compact. 
and let ![M\subseteq [0,1]](https://s0.wp.com/latex.php?latex=M%5Csubseteq+%5B0%2C1%5D&bg=ffffff&fg=000000&s=0&c=20201002)
be the set of all points ![a\in[0,1]](https://s0.wp.com/latex.php?latex=a%5Cin%5B0%2C1%5D&bg=ffffff&fg=000000&s=0&c=20201002)
such that the subsegment ![[0,a]](https://s0.wp.com/latex.php?latex=%5B0%2Ca%5D&bg=ffffff&fg=000000&s=0&c=20201002)
admits a finite subcovering selected from 
contains some positive number. Denote by 
the supremum of points in 
. We claim that 
. Indeed, if 
, then 
for some open set 
. But since 
would still be in ![[0,1]](https://s0.wp.com/latex.php?latex=%5B0%2C1%5D&bg=ffffff&fg=000000&s=0&c=20201002)
admits a finite subcovering selectable from 
is not compact. To see this, let us enumerate all points of 
by natural numbers, 
(this is possible, since 
by the interval (open ball) of radius 
centered at this point. This infinite covering does not admit a finite subcovering of 
which is less than 
, so at least one point of 
be topological spaces and 
is compact in 
be an arbitrary open covering of 
. By continuity of 
give a finite covering of 
) and closed (
). Q.E.D. 
added instead. We can introduce a “perverse topology” on this set 
by declaring that the open sets are the open sets in the former 
which (check it at home) are consistent with claiming that 
is a topological space. 
and 
, cannot be separated by disjoint open sets: any two sets 
open in the topology of 
. This means that the topology of 
. 
and taking values in another space 
is “too narrow” to talk in proper geometric terms. 
sufficiently close to each other, their images 
and $b=f(a)$ in 
. This function should be far from arbitrary: 
for any pair of points; the distance is zero if and only if the two points coincide, 
, that is, the distance is a symmetric function;
we have 
satisfies all these axioms. (Don’t think that this is the only function with such properties! The function 
also satisfies all of them.) Yet in 
there is a whole family of distance functions, ![\textrm{dist}(x,y)=\sqrt[p]{\sum_{i=1}^n(x_i-y_i)^p}](https://s0.wp.com/latex.php?latex=%5Ctextrm%7Bdist%7D%28x%2Cy%29%3D%5Csqrt%5Bp%5D%7B%5Csum_%7Bi%3D1%7D%5En%28x_i-y_i%29%5Ep%7D&bg=ffffff&fg=000000&s=0&c=20201002)
for any positive 
. The case 
appears as the limit case when 
grows to infinity, the case 
corresponds to the usual Euclidean metric. 
we will call open balls centered at 
for any of the above distance functions on 
for all 
(invariance by translations). Not all functions satisfying the above three properties possess this invariance. 
. 
is trivially valid for any function and the axiom of distance! Besides, it is not clear, for which pairs 
this claim must hold.
can be called 
-small, with two positive numbers 
, getting the implication
.
that involves four “free variables”, 
. What shall we do with them? There are two options, either to tie down each variable by one of the two quantifiers $\forall, exists$ or to designate them as a “parameter” that has to be specified in advance. 
. They clearly play a symmetric role, so it would be natural to assign the same quantifier in front of each of them, and this quantifier is obviously “for all”, 
(warning: see below). As for the two remaining “scale variables” 
, we can play around with the two quantifiers 
and 
, where 
are independently either 
or 
, and placed in a different order: altogether this yields 8 various possible combinations (some of them identical). 
.
-closeness (resp., 
in the target space one can find the proximity measure 
are 
. Then if 
, then for any fixed 
will eventually exceed any given 
is large enough, so the quadratic function, the nicest of all nonlinear function, turns out to be not uniformly continuous on its natural domain. This happens because the natural domain 
.
and considering it as a parameter. The corresponding definition (stripped of the word “uniformly“, looks as follows. 
such that 
its image 
. 
that depends on two logical “free variables” 
converges” takes the form involving the horrible four (!) quantifiers:
, where the claim 
, involving the number 
Sergei Yakovenko's blog: on Math and Teaching 
