Separable space definition and applications

[Somefantastik]

I read in my metric spaces book that a separable space is that which has a countable, dense subset. This definition has no intuitive meaning to me. I'm able to show if a space is dense or not, and I think I can show a space is countable. But, I'm missing the "so what?!"

I would like to understand this concept better. Perhaps the root of the problem is my lack of understanding of the definition of dense. I know dense means the closure of the space is equal to the space (the space and its derived set). I know if a set A is dense in B, then for every element a of A, there exists a sequence bn in B such that bn approximates a (can't say bn converges to a since a not in B).

Am I missing something about the concept of dense maybe?

What sorts of fun things does one do with a separable space? What does a nonseparable space imply?

Just a note - I am not familiar with topology, only real analysis and metric spaces. The way my classes were structured, we seemed to skip most concepts involving open sets , so if there's a way to help me understand without talking about open sets or open covers, that would be really appreciated.

 

[DonAntonio]

I read in my metric spaces book that a separable space is that which has a countable, dense subset. This definition has no intuitive meaning to me. I'm able to show if a space is dense or not, and I think I can show a space is countable. But, I'm missing the "so what?!"

I would like to understand this concept better. Perhaps the root of the problem is my lack of understanding of the definition of dense. I know dense means the closure of the space is equal to the space (the space and its derived set). I know if a set A is dense in B, then for every element a of A, there exists a sequence bn in B such that bn approximates a (can't say bn converges to a since a not in B).



*** Here it is exactly the other way around: A dense in B means that for any element in B there exists a sequence in A that converges to that element. ***


Am I missing something about the concept of dense maybe?


*** Perhaps. Another equivalent definition of dense set, which for me is way easier to conceptualize, is the following:

A set A in a topological space X (take this to be a metric space, if you prefer) is dense in X iff $A\cap U\neq \emptyset$ for any

open non-empty set $U\subset X$

DonAntonio ***


What sorts of fun things does one do with a separable space? What does a nonseparable space imply?

Just a note - I am not familiar with topology, only real analysis and metric spaces. The way my classes were structured, we seemed to skip most concepts involving open sets , so if there's a way to help me understand without talking about open sets or open covers, that would be really appreciated.

...

 

[xaos]

if the space has only an uncountable basis, you can't build metric space structure onto it. so one way of getting a countable basis is in finding a countable dense subset to attach it to.