Hausdorff spaces

In the preceding post (coffee break 8-2), a topological space is defined by open subsets.

Putting simply, a topological space is a collection of open sets.

However, as this definition is too general, some problems will occur.

First of all, when $a_n\rightarrow x$  and $a_n\rightarrow y$ , $x=y$  may not be proved.

$a_n\rightarrow x$  means that, for an $N<\infty$ , if $N<n$ , all $a_n$ is included

in the neighborhood of $x$ , where the neighborhood of $x$  is an open subset
in the topological space which includes $x$ .

In the definition of a topological space, we may not say the neighborhood of $x$ and $y$  is
same or different.

Therefore, we prepare the topological space such that, if two elements $x$  and $y$  is different,

each neighborhood of elements is pairwise disjoint. Such a space is called Hausdorff space,
a separable space, or $T2$ space.

coffee break 8-2 (topological spaces)

Axiomatic structure is not just only in probability theory.

All fields in mathematics is constructed by axiomatic structure.

It is no problems to say that Mathematics itself has axiomatic structure.

 

Except Axioms which are accepted by no proofs, as definitions, propositions,

theorems, and corollaries are all needed a proof of validity or consistency,

at this point, all is same. 

 

Axioms are assertions in which no one has questions.

However, we are not always able to address mathematical issues from using axioms,

because the explanations are very long.

 

Then, we usually begin with basic definitions which have been consistently gotten by axioms.

For example, a topological space is defined as follows.

 

A topological space $T$  is a set which has open subsets and satisfies the following conditions.

(1) $T$  itself (and empty set ) is open.

(2) the intersection of two open subsets in $T$  is open.

(3) the union of any open subsets in $T$  is open.

 

As you know, a topological space is a generalized distance space.

However, there are not distance functions in the definition.

It is necessary for us to understand that open sets can take a role of distance functions.

I think that such a beginning makes the gate of mathematics narrow and narrow.

coffee break 8 (probability measures)

It is well known that one of the main ideas underlying in Black-Scholes’ formula is the risk neutral probability.  

As the risk neutral probability might be gotten by solving given equations, it is usually explained in Finance that the risk neutral probability is artificial.

However, in mathematics, every probability measures $P$ is artificial. Because those may just satisfy the only following definitions.

(1)$P(\Omega)=1,\quad P(\phi)=0$

(2)$0\leq P(A) \leq 1\quad (A\subset \Omega)$
(3)$P(\cup A_i)=\sum P(A_i)$

Such the axiomatic probability theory was founded by Kolmogorov in the 1930s.

Many mathematicians accepted gladly the new paradigm.

Then, the axiomatic structure based upon Lebesgue theory gave birth to big progress in the probability theory.

However, on the other hand, as Kiyoshi Ito has once noted, the structure has made an enormous gap between those who are familiar with mathematics and those who are not.