Difference Between Subbasis and Basis in Topology

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In summary, subbasis and basis are concepts in topology that help define a topology on a given set. A subbasis is a collection of subsets of the set that can be used to generate a topology, while a basis is a collection of subsets that satisfy certain properties and can be used to define a topology on a set. Both subbasis and basis play important roles in defining topologies and can be used interchangeably in certain cases.
  • #1
tomboi03
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This is probably a stupid question.. but

Can someone tell me the difference between subbasis and basis.. in topology?? I know the definitions...

So Subbasis is defined to be the collection T of all unions of finite intersections of elements of S (subbasis)

sooo... S is pretty much a topology on X which is a collection of subsets of X whose union equals X.

Basis, however... is
If X is a set, basis on X is a collection B of subsets of X (basis elements) s.t.
1. for each x [tex]\in[/tex] X, there is at least one basis element B containing x.
2. If x 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.

Right? So pretty much... A subset U of X is said to be open in X if for each x [tex]\in[/tex] U, there is a basis element B [tex]\in[/tex] [tex]B[/tex] such that x [tex]\in[/tex] B and B [tex]\subset[/tex] U.

But I'm still not understanding this quite... so well..

Can someone explain this to me??


Thank You!
 
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  • #2
The topology generated by a subbasis is the topology generated by the basis of all finite intersections of subbasis elements. It seems like you just need an example, though, so here's one.

A basis for the standard topology of R is the collection of all open intervals. A subbasis for R is the collection [tex]\{(a,\infty):a\in \mathbb{R}\} \cup \{(-\infty,b):b\in \mathbb{R}\}[/tex]. The reason that this is a subbasis for R is because finite intersections of elements in this set are precisely the basis elements of R. For instance, we have [tex](a,b) = (-\infty,b)\cap (a,\infty) [/tex]. Subbases are often important because they offer an easier way of expressing a topology. In the previous example, there are many fewer sets in the sub-basis than there are in the basis.
 
  • #3
I think there is a distiction that can help you avoid confusion.

If you have a set [tex]X[/tex] and NO topology on it
every [tex]S[/tex] subset of the set [tex]P(X)[/tex] of the parts of [tex]X[/tex]
can be taken and you can generate e topology that has [tex]S[/tex] as subbasis.

If you have a set [tex]X[/tex] and NO topology on it then
if you have a set [tex]B \subset P(X)[/tex]
that satisfies the 2 property 1. ans 2. you mentioned you can generate a topology on
[tex]X[/tex] that has [tex]B[/tex] has basis.


Now if you have a set [tex]X[/tex] and HAVE a topology [tex]T[/tex] on it
a set [tex]S \subset P(X)[/tex] is a subbasis of [tex](X,T)[/tex] if and only if (by definition)
for every open set [tex]A[/tex] and for every [tex]a \in A[/tex] exists
[tex]S_1, \ldots, S_n \in S[/tex] such that
[tex]a \in S_1 \cap \cdots S_n \subset A[/tex].


Now if you have a set [tex]X[/tex] and HAVE a topology [tex]T[/tex] on it
a set [tex]B \subset P(X)[/tex] is a basis of [tex](X,T)[/tex] if and only if (by definition)
for every open set [tex]A[/tex] and for every [tex]a \in A[/tex] exists
[tex]B_1 \in B[/tex] such that
[tex]a \in B_1 \subset A[/tex].



There is a slight difference to understand.
Please read it carefully and think on it and you'll get the concept.
Hope this helps.
 

Related to Difference Between Subbasis and Basis in Topology

What is a subbasis in topology?

A subbasis in topology is a collection of subsets of a topological space that forms a basis for the topology. These subsets are usually used to define the open sets in the topology, and they can be combined to create a basis for the entire topology.

How is a subbasis different from a basis in topology?

A subbasis is a smaller collection of subsets compared to a basis. While a basis contains enough sets to form a topology, a subbasis may not be enough and may require additional sets to create a basis. Additionally, a subbasis does not necessarily need to be closed under finite intersections, unlike a basis.

Can a subbasis generate multiple topologies?

Yes, a subbasis can generate multiple topologies. This is because the choice of which subsets from the subbasis to include in the basis can vary, resulting in different topologies. However, the generated topologies will always have the subbasis as a subset.

What is the significance of subbasis in topology?

Subbasis is important because it allows for a more efficient way of defining a topology. Instead of having to list out all possible open sets, a smaller collection of subsets can be used. This is particularly useful when dealing with topologies on infinite spaces.

Can a subbasis be used to define a topology on any set?

No, a subbasis can only be used to define a topology on a set if the collection of subsets satisfies certain conditions. These include being non-empty, covering the entire set, and being closed under finite intersections. If these conditions are not met, the subbasis cannot be used to define a topology.

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