The Cartesian product theorem for dimension 0

In summary, the Cartesian product theorem for dimension 0 states that the Cartesian product of two sets with 0 dimensions is also a set with 0 dimensions. This is different from the Cartesian product theorem for higher dimensions, where the product is a set of ordered pairs. The significance of this theorem lies in its fundamental role in set theory and its practical applications. It can also be applied to sets with any number of elements, as long as they have 0 dimensions. Additionally, the Cartesian product theorem for dimension 0 is closely related to the concept of the empty set, as the product of an empty set with any other set is always the empty set.
  • #1
hedipaldi
210
0
The cartesian product ∏X = Xi of a countable family {Xi} of regular spaces is zero-dimensional
i f and only i f all spaces Xi , are zero-dimensional.
I wonder if the countability assumption is just to ensure the regularity of the product space ,or it is crucial for the clopen basis.
Thank's
 
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  • #2
Products of an arbitrary number of regular spaces are always regular, so the problem isn't there. Did you check the proof?? Where did they use countable?
What book is this anyway?
 
  • #3
The proof seems to hold for uncountable product.The proof is attached.
 

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Related to The Cartesian product theorem for dimension 0

1. What is the Cartesian product theorem for dimension 0?

The Cartesian product theorem for dimension 0 states that the Cartesian product of two sets with 0 dimensions (i.e. two single points) is also a set with 0 dimensions (i.e. a single point).

2. How is the Cartesian product theorem for dimension 0 different from the Cartesian product theorem for higher dimensions?

In higher dimensions, the Cartesian product of two sets is a set of ordered pairs, while in dimension 0, it is simply a single point. This is because in dimension 0, there is only one possible combination of elements from the two sets, while in higher dimensions, there are multiple possible combinations.

3. What is the significance of the Cartesian product theorem for dimension 0?

The Cartesian product theorem for dimension 0 is a fundamental concept in set theory that helps explain the properties of the Cartesian product and lays the groundwork for understanding the Cartesian product in higher dimensions. It also has practical applications in fields such as computer science and statistics.

4. Can the Cartesian product theorem for dimension 0 be applied to sets with more than two elements?

Yes, the theorem can be applied to sets with any number of elements, as long as they each have 0 dimensions. This means that the Cartesian product of any number of single points will always result in a single point in dimension 0.

5. How does the Cartesian product theorem for dimension 0 relate to the concept of the empty set?

The Cartesian product of an empty set with any other set (including another empty set) is always the empty set. This is because the empty set has 0 dimensions, and according to the Cartesian product theorem for dimension 0, the product of two sets with 0 dimensions is also a set with 0 dimensions (i.e. the empty set).

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