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.