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Definition of a subbasis of a topology

  1. Nov 25, 2013 #1
    One of the definitions of a subbasis ##\mathcal{S}## of a set ##X## is that it covers ##X##. Then the collection of all unions of finite intersections of elements of ##\mathcal{S}## make up a topology ##\mathcal{T}## on ##X##. That means the collection of all finite intersections of elements of ##\mathcal{S}## is a basis ##\mathcal{B}## for the topology ##\mathcal{T}##.

    But one of the defining characteristics of a basis is that it also must cover ##X##, although if the subbasis is the collection of all singletons in ##X##, which definitely covers ##X##, then the basis ##\mathcal{B}## would have only the empty set; wouldn't it?
     
  2. jcsd
  3. Nov 25, 2013 #2
    If [itex]\mathcal S = \{ \{x\}: \enspace x\in X\}[/itex] as you describe, then the set of all finite intersections of members of [itex]\mathcal S[/itex] is just [itex]\mathcal B = \mathcal S \cup \{\emptyset, X\}[/itex]. This is a basis for the topology [tex]\mathcal T = \left\{ \bigcup \hat{\mathcal B}: \enspace \hat{\mathcal B} \subseteq \mathcal B \right\}= \{A: \enspace A\subseteq X \}[/tex] which some call the discrete topology.
     
  4. Nov 25, 2013 #3
    How does the intersection of singletons of a set give you the whole set? I can see that they give you the empty set, but not the whole set itself. Or are you just throwing the whole set in to complete the basis??
     
  5. Nov 25, 2013 #4
    It doesn't give ##X##. But the process to form a topology given a subbasis ##\mathcal{S}## is the following:
    1) First adjoin ##\emptyset## and ##X##.
    2) Take all finite intersections
    3) Take all unions

    So this is why he had the set ##X##, since you need to adjoin it according to (1).

    However, you seem to have a bit of another definition of a subbasis. You demand that a subbasis covers ##X##. This is not the standard definition, I believe. But if you follow your definition than the steps are:
    1) Adjoin ##\emptyset##
    2) Take all finite intersections
    3) Take all unions.
     
  6. Nov 25, 2013 #5
    It's not that important, but I was using the convention that the intersection of no sets is the whole space. i.e. Given a universe [itex]X[/itex], for any collection [itex]\mathcal A \subseteq 2^X[/itex] of sets, one common definition of the intersection is [itex]\bigcap \mathcal A:= \{x\in X: \enspace x\in A \text{ for every } A\in\mathcal A\}.[/itex] If this is the definition you like, then [itex]\bigcap\emptyset=X.[/itex] Other people adopt the convention that "[itex]\bigcap \emptyset [/itex]" is just undefined.
     
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