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Basis topology

  1. Mar 6, 2009 #1
    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 [tex]B[/tex] of subsets of X (called basis elements) satisfying the following properties.

    1. For each x[tex]\in[/tex]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[tex]\subset[/tex]B1[tex]\cap[/tex]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[tex]\in[/tex]U, there is a basis element B [tex]\in[/tex] [tex]B[/tex] and B[tex]\subset[/tex]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!
     
  2. jcsd
  3. Mar 6, 2009 #2
    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?
     
  4. Mar 6, 2009 #3
    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?
     
  5. Mar 6, 2009 #4
    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 [tex]\cup[/tex] and [tex]\bigcup[/tex] denote the union of two sets and the union of a family of sets respectively.
     
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