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I Definition of topology

  1. Feb 28, 2017 #1
    Hello! I just started reading an introductory book about topology and I got a bit confused from the definition. One of the condition for a topological space is that if ##\tau## is a collection of subsets of X, we have {##U_\alpha | \alpha \in I##} implies ##\cup_{\alpha \in I} U_\alpha \in \tau ##. I assume this means that for any 2 sets in ##\tau## their union is also in ##\tau##. But I really don't understand the notation. What does {##U_\alpha | \alpha \in I##} mean? And how is it related to ##\tau##? And what is I? There is nothing before this, to define "I" and I found this definition in different books, so I assume i am missing something here. Thank you!
     
  2. jcsd
  3. Feb 28, 2017 #2

    fresh_42

    Staff: Mentor

    A topological space ##(X,\tau)## is a set ##X## together with a set ##\tau## of subsets ##U_\alpha \subseteq X\, , \,\alpha \in I##. This means any number of subsets. Next there are some requirements for this set of sets ##\tau = \{U_\alpha \subseteq X\, | \,\alpha \in I\}##.
    The empty set ##\emptyset## as well as the entire set ##X## have to be elements of ##\tau##.
    Any finite intersection of ##U_\alpha \in \tau## must also be an element of ##\tau##.
    Any arbitrary union of ##U_\alpha \in \tau## must also be an element of ##\tau##.
    If these requirements hold, then ##\tau## is called a topology on ##X##, which means the elements ##U_\alpha## of ##\tau## are the subsets of ##X## we call open.

    In short: A topology on ##X## is the definition ##\tau## of sets, which we call open sets.
    As long as the conditions above hold, we're free to choose any subsets of ##X## as open. That's why spaces can carry more than one topology.
     
  4. Feb 28, 2017 #3
    Thank you for your reply. It makes more sense now. So, just to make sure I understand, for example in R, we can define ##\emptyset## and R as the open sets in R and in this topology, by definition there are only 2 open sets? And the fact that we usually consider any interval (a,b) to be an open set in R is just a particular choice of topology on R?
     
  5. Feb 28, 2017 #4

    fresh_42

    Staff: Mentor

    Yes.
    You probably see, that ##\{\emptyset,\mathbb{R}\}## isn't very interesting as a topology. It's the weakest or coarsest topology you can choose. The strongest or finest topology would be to declare for every single point ##p## the set ##\{p\}## as open set, which makes all subsets open (and simultaneously closed).

    The "usual" topology defined by open intervals, is the topology, that comes from the Euclidean norm of ##\mathbb{R}## which gives us a tool to measure distances and ##"<"## conditions define something open, ##"\geq"## conditions something closed.
     
  6. Mar 8, 2017 #5

    Stephen Tashi

    User Avatar
    Science Advisor

    For example, suppose we are in the "usual" topology on the real line where open intervals are examples of open sets. let ##I## denote the set of real numbers that are not rational numbers of the form ##\frac{k}{2^n}## where ##k## and ##n## are positive integers. For each ##r \in I## define ##U_r## to be the open interval ##( r - e^{-|r|}, r + e^{-|r|} )##. The union ##S## of all the sets ##U_r## cannot be represented as an infinite series of unions of the form ##U_{r_1} \cup U_{r_2} \cup U_{r_3} \cup ...## because such a representation assumes the indices are the countable infinity of integers, which isn't sufficient to index the uncountable number of sets we are dealing with. So the notation ##S = \cup_{\alpha \in I} U_{\alpha}## is used to indicate the uncountable union.

    This is illustrates the distinction between an "arbitrary" union and a "countably infinite union".
     
    Last edited: Mar 8, 2017
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