Find a topological space which does not have a countable basis

In summary, the conversation discusses the task of finding a topological space without a countable basis. The conversation mentions trying to use a space of cantor-like numbers or creating a very large underlying set to achieve this. It is noted that there is complete freedom in choosing both the set and the topology for this task.
  • #1
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Homework Statement


Find a topological space which does not have a countable basis.


Homework Equations


Definition of basis : A collection of subsets which satisfy:
(i) union of every set equals the whole set
(ii) any element from an intersection of two subsets is contained in another subset which is itself contained in the intersection


The Attempt at a Solution


I thought I had it nailed, but the solution i came up with was actually an uncountable basis for the standard euclidean topology, which can also have countable bases. So i need to think of a space which cannot have a countable basis.

I started thinking about a space of cantor-like numbers (infinite strings of 1's and 0's) but this is still a subset of the reals! I am stuck please any hints would be a blessing.
 
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  • #2
You have complete freedom to choose both the set and the topology you give it. So one strategy you could try is to make your underlying set very large -- say, a large number of copies of some familiar space. Alternatively you could try starting with a familiar set like [tex]\mathbb{R}[/tex], but giving it very many open sets, many more than the standard topology has.
 

1. What is a topological space?

A topological space is a set of points with a set of open subsets, which satisfies certain axioms such as closure under finite intersections and arbitrary unions.

2. What is a countable basis?

A countable basis for a topological space is a collection of open subsets that can be used to generate all other open subsets in the space. In other words, any open subset in the space can be written as a union of elements in the countable basis.

3. Why is it important for a topological space to have a countable basis?

A countable basis allows for a more efficient and manageable way to describe the open sets in a topological space. It also allows for easier analysis and understanding of the space's properties.

4. How do you construct a topological space without a countable basis?

One way to construct a topological space without a countable basis is by using the cocountable topology on an uncountable set. In this topology, the open sets are the complements of countable sets. This space does not have a countable basis because any countable collection of open sets will not be able to generate all other open sets in the space.

5. What are some real-world examples of topological spaces without a countable basis?

One example is the space of all real numbers with the cocountable topology, as mentioned in the previous answer. Another example is the space of all continuous functions on a given interval, where the open sets are defined by pointwise convergence. This space does not have a countable basis because any countable collection of open sets will not be able to generate all other open sets in the space.

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