Product Space: Example of Unrestricted Open Set Not Open in Product Topology

In summary, the conversation discusses the concept of the product topology and how an unrestricted product of open sets in the coordinate spaces may not necessarily be open in the product topology. An example is given using the Cantor set, which is not discrete but can be represented as a product of countably many copies of the discrete space {0,1}. The conversation also explores the idea of taking the product of infinitely many open sets and how it may not result in an open set in the product topology. An example is given using the real line as the product space and (0,1) as the open sets. The conversation concludes by discussing how the singleton in the product space is not open due to the properties of the product topology.
  • #1
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Hi,
I'm starting to read an introduction to topology book.
In the chapter about the topology of the product space (product of topological spaces), it says: "It should be clearly understood that an unrestricted product of open sets in the coordinate spaces need not be open in the product topology".

Can anybody please give me an example of this?
Thank you
 
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  • #2
Wikipedia mentions that the Cantor set is homeomorphic to a product of countably many copies of the discrete space {0,1}.

The Cantor set is not discrete. But note that every subset of {0,1}N is a union of set-theoretic products of open subsets of {0,1}.
 
  • #3
Thanks for the reply Hurkyl.

This is what I was thinking: We get into the quoted situation when the amount of spaces in the product is infinite. If we take the product of infiinitely many open sets smaller than the coordinate spaces, then we have infinitely many projections smaller than the coordinate spaces. But then this resulting set wouldn't be in the product topology because only a finite amount of projections of a set in the product topology can be different than the coordinate spaces.
So for example if our product space is the product of the real line infinitely many times, then taking the product of (0,1) infinitely many times does not result in an open set in the product topology.
Is this right?
 
  • #4
Our space C is the infinite product of {0,1}. In the space {0,1}, the singleton {0} is open. But the product of {0} in every factor (which gives us a singleton in C) is not open in C.
 
  • #5
Thanks g_edgar,
The singleton in C is not open because of what I explained in my previous reply, right?
 

1. What is product space and how is it defined?

Product space is a mathematical concept used in topology to describe the set of all possible combinations of elements from two or more given sets. It is defined as the Cartesian product of the given sets, where each element in the product space is an ordered pair consisting of one element from each of the given sets.

2. How is the product topology defined and how does it relate to product space?

The product topology is a topology defined on the product space, which is a collection of subsets that satisfy certain properties. It is defined as the topology generated by the basis elements, which are the open sets of the individual spaces in the product. The product topology is used to study the properties of the product space, such as continuity and convergence.

3. Can an unrestricted open set in product topology not be open in product space?

Yes, it is possible for an unrestricted open set in product topology to not be open in product space. This occurs when the open set in the product topology is a product of open sets in the individual spaces, but the resulting set in the product space is not open. This can happen when the open sets in the individual spaces do not intersect in the product space.

4. What is an example of an unrestricted open set not being open in product topology?

An example of an unrestricted open set not being open in product topology is the set {(x,y) | x > 0 and y > 0} in R^2, where R represents the set of real numbers. This set is open in the product topology, as it is a product of the open sets (0,∞) in the individual spaces. However, the resulting set is not open in the product space, as it does not contain points on the x or y-axis.

5. How is the concept of product space and product topology used in scientific research?

The concept of product space and product topology is used in various fields of science, such as physics, chemistry, and biology. In physics, it is used to study the properties of multi-particle systems and in chemistry, it is used to analyze the properties of chemical compounds. In biology, it is used to study the interactions between different biological systems. Additionally, the product topology is also used in data analysis and machine learning to model and analyze complex data sets.

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