/sci/ - Science & Math

[Anonymous No.2093901]

/sci/,
What do I need to completely formally describe the empty space?
I think I need a basis, a metric, and a field. I think the basis will be the empty set.

[Anonymous No.2093905]

Whoa whoa whoa.
What type of space?

A set with a metric is called a metric space.
I don't know any type of object that is specified by a set, a metric and a field.

[Anonymous No.2093912]

>>2093905
My understanding of spaces is that they are defined for a basis over a field by a metric.

Euclidean spaces are equipped with the Euclidean metric, a basis from the real numbers over the field of the real numbers.

I can't tell you what type of space because I don't know what it would be at the moment; hence, I want to construct it.

What I know about it is that the number of elements in it's basis is zero.

[Anonymous No.2093937]

>>2093912
The only use of the "word" basis that I know of is for a vector space.

A vector space needs to be over a particular field. It doesn't matter which field you pick, you can still have an empty vector space over it. Let's say the field is called F, with identity elements 0_F and 1_F and operations *_F and +_F.

Then the empty vector space over F is a set containing a single element 0_V equipped with the only possible operations for vector addition and scalar multiplication (i.e. everything maps to 0_V).

You can check that this is indeed a vector space over F, with identity 0_F. The only basis for this vector space is the empty set.

[Anonymous No.2093939]

Sorry I just realised, that would probably be called the TRIVIAL vector space over F, not the "empty" vector space over F.

There can't be a vector space with literally no elements, because one of the axioms for a vector space requires an additive identity.

[Anonymous No.2093997]

>>2093939
We need a weaker notion of a space then what about a topological space?