Clarifying Topology Basics: What is U?

[tomboi03]

This is a very simple question...

Because I'm not very good at these... notations... I feel like I need a clarification on what this means..

if X is a set, a basis for a topology on X is a collection $$B$$ of subsets of X (called basis elements) satisfying the following properties.

1. For each x$$\in$$X, there is at least one basis element B containing x.

2. If belongs to the intersection of two basis elements B₁ and B₂, then there is a basis element B₃ containing x such that B₃$$\subset$$B₁$$\cap$$B₂.

If B satisfies these two conditions, then we define the topology T generated by B as follows: A subset U of X is said to be open in X (That is, to be an element of T) if for each x$$\in$$U, there is a basis element B $$\in$$ $$B$$ and B$$\subset$$U. Note that each basis element is itself an element of T.

okay... i know what basis topology is... by reading all of these.. but i want a clarification on what this... U is... is it the whole set? like the Union? or what? or is it just a variable that they define?

I just want a clarification..

Thank you!

 

[yyat]

U is any subset of X.

What this means is the following: You want to define a topology on X, so you have to say which subsets are open. So if you are given any subset U of X, here is how you can tell if it is open: if for each x in U... and so on.

Does that help?

 

[tomboi03]

It helps but...

Can i ask when U is used for the whole set?

Because doesn't the subject topology use U as the whole set?

 

[yyat]

It helps but...

Can i ask when U is used for the whole set?

Because doesn't the subject topology use U as the whole set?

What do you mean by "the whole set". If X is a topological space, then the whole space is X, if you call the topological space U, then U is the whole space.

The symbols $$\cup$$ and $$\bigcup$$ denote the union of two sets and the union of a family of sets respectively.