Hausdorffness of the product topology

In summary, the conversation discusses the Hausdorffness of the product topology and whether or not the product of an infinite number of Hausdorff spaces is always Hausdorff. The speaker mentions that the product topology has a basis consisting of products of open sets, and that it has been proven that the product of an infinite number of Hausdorff spaces is Hausdorff. However, the speaker brings up the question of whether or not the converse is true. They also mention that the product must not be empty for the converse to hold. The conversation also touches on the issue of finding two neighborhoods that do not intersect in the product space.
  • #1
quasar987
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[SOLVED] Hausdorffness of the product topology

Is it me, or is the product of an infinite number of Hausdorff spaces never Hausdorff?

Recall that the product topology on

[tex]\Pi_{i\in I}X_i[/tex]

has for a basis the products of open sets

[tex]\Pi_{i\in I}O_i[/tex]

where all but finitely many of those O_i are not the whole X_i.

---

say I is countable for simplicity and consider x=(x1,...) and y=(y1,...) two distinct points in the product space. I don't see how we can find two ngbh of x and y that do not intersect!
 
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  • #2
Hausdorff, given either the product topology or box topology. This has been proven here before. The converse is true as well (though mathwonk remarked that the product must not be empty).
 
  • #3
I found the thread thx.
 
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  • #4
But it seems to deal only with the converse.

Given any 2 open set of the basis containing x and y resp., since infinitely many O_i are X_i in both ngbh, they will always have a huge intersection. Is this not the case?
 
  • #5
Ask yourself this question: Is there anything in [itex]\emptyset \times X \times X \times \cdots[/itex]?
 

Related to Hausdorffness of the product topology

1. What is the definition of Hausdorffness in the context of topology?

Hausdorffness, also known as the Hausdorff property or the separation axiom, is a fundamental concept in topology that describes the level of separation between points in a topological space. In a Hausdorff space, any two distinct points can be separated by disjoint open sets.

2. How does Hausdorffness of a topological space impact its properties?

The Hausdorff property is an important condition for many topological spaces because it guarantees certain desirable properties such as uniqueness of limits and continuity of functions. Additionally, Hausdorff spaces tend to have more structure and are easier to work with in many mathematical contexts.

3. Can the product of two Hausdorff spaces be non-Hausdorff?

Yes, it is possible for the product of two Hausdorff spaces to be non-Hausdorff. This occurs when the product topology on the Cartesian product of the two spaces does not satisfy the Hausdorff condition. In other words, there exist two points that cannot be separated by disjoint open sets in the product space.

4. What is the significance of Hausdorffness in the product topology?

In the context of the product topology, Hausdorffness ensures that the product space has the same desirable properties as its individual component spaces. It guarantees that the product of two Hausdorff spaces is also Hausdorff, which is important for preserving the structure and properties of the original spaces.

5. How is the Hausdorffness of a product topology determined?

The Hausdorffness of a product topology can be determined by checking if the product of any two distinct points in the Cartesian product space can be separated by disjoint open sets. If this condition is satisfied, then the product topology is Hausdorff. If not, then the product topology is not Hausdorff.

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