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mathboy
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I know that a product of Hausdorff spaces is Hausdorff. Is the converse also true? That is, if A_1 x A_2 x A_3 x... is Hausdorff, then is each A_i Hausdorff?
PxH is empty if P is empty because if f belonged to PxH, then f(1) would belong to P and f(2) would belong to H. But f(1) cannot belong to P because P is empty. So there is no such f.Singularity said:If say, P is the empty set with the discrete topology and H is another hausdorff space,
will PxH be homeomorphic to H?
Singularity said:Can a function "belong" to a space? I have never heard of this. Also, your argument only has one component, when it seems like it should have two. Am I missing something fundamental here?
Thanks :)
A product of Hausdorff spaces refers to the topological space obtained by taking the Cartesian product of two or more Hausdorff spaces. It is a common construction in topology that allows us to study the properties of multiple spaces simultaneously.
A product of Hausdorff spaces inherits the Hausdorff property, meaning that for any two distinct points in the product, there exist disjoint open sets containing each point. It also has the product topology, which is the smallest topology that makes all the projection maps continuous.
A product of Hausdorff spaces is useful for studying the properties of Cartesian products, which appear naturally in many mathematical objects. It is also a helpful tool in proving results about topological spaces, such as Tychonoff's theorem which states that the product of compact spaces is compact.
Yes, a product of Hausdorff spaces can be non-Hausdorff. This can happen if one or more of the individual spaces in the product is non-Hausdorff. However, if all the spaces in the product are Hausdorff, then the product will also be Hausdorff.
The product of Hausdorff spaces and the product of topological spaces are closely related, but they are not the same. The product of topological spaces only requires the projection maps to be continuous, while the product of Hausdorff spaces has the additional requirement of being Hausdorff. In other words, the product of Hausdorff spaces is a stricter version of the product of topological spaces.