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What the following statement exactly mean?

  1. Mar 18, 2014 #1
    what the following statement exactly mean :"A topological space is a set M with a distinguished collection of subsets,to be called the open sets".and I don't understand the meaning of "distinguished collection ".it's better have some example.thanks
     
  2. jcsd
  3. Mar 19, 2014 #2

    micromass

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    It means that in order to give a topological space, you need to give two pieces of information:

    1) First, you need to give an underlying set ##X##
    2) Secondly, you need to give a set ##\mathcal{T}## which consist of subsets of ##X##. This set ##\mathcal{T}## is called the topology. It can be completely arbitrary, but it does need to satisfy the axioms of a topological space.

    For example,
    1) I give you the set ##X=\mathbb{R}##
    2) I give the topology ##\mathcal{T} = \{\emptyset,\mathbb{R}\}##.
    These two pieces of data specify a topological space, called an indiscrete space.

    Other example:
    1) I give you the set ##X=\{0,1\}##
    2) I give the topology ##\mathcal{T} = \{\emptyset,\{0,1\},\{0\}\}##
    These two pieces of data specify the Sierpinski topological space.

    Obviously, not everything will be a topological space. For example
    1) I give you ##X=\{0,1,2\}##
    2) I give you ##\mathcal{T} = \{\emptyset, \{0,1,2\}, \{0,1\}, \{1,2\}\}##
    These are again two pieces of data, but they do not specify a topological space because ##\mathcal{T}## does not satisfy the axioms. Indeed, ##\{0,1\}\cap\{1,2\} = \{1\}## is not in ##\mathcal{T}##.
     
  4. Mar 19, 2014 #3
    both “a distinguished collection of subsets”and “to be called the open sets” refer to the set T ?
     
  5. Mar 19, 2014 #4

    micromass

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    Yes, ##\mathcal{T}## is the "distinguished collection of subsets". Any element of ##\mathcal{T}## is by definition an open set.
     
  6. Mar 19, 2014 #5
    now i understand,thanks
     
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