Proving Equivalence of Standard and Basis-Generated Topologies on RxR

In summary, the conversation discusses the equivalence of the standard topology on RxR and the topology generated by the basis consisting of open disks. The conversation also explains the definitions of a basis and the topology generated by a basis, as well as how the product topology in RxR makes "open rectangles" open sets. The conversation concludes by discussing the process of finding open discs inside rectangles and how it relates to the topic at hand.
  • #1
hello12154
2
0
I am having trouble proving this statement. Please help as I am trying to study for my exam, which is tomorrow

Prove that the standard topology on RxR is equivalent to the one generated by the basis consisting of open disks.

Thanks :)
 
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  • #2
What are you having trouble understanding? What is the natural metric topology of R? This is essentially the same that is put on RxR, pointwise. Can you see how this coincides with open disks? It is not a hard problem.
 
  • #3
What is the topology generated by the basis consisting of open disks?
 
  • #4
What is the definition of a basis? What is the definition of the topology generated by a basis? Read carefully the definition and think if you completely understand every word and every statement there. Then close your book, write the definitions yourself on a blank piece of paper, as precisely as you can. Then open the book and compare - did you skip or added something? Once this is done, it will be easier to discuss the details and the difficulties you still may have.
 
  • #5
Suppose a U is an open set in R2 (consider the usual euclidean topology - we are drawing a distinction between R2 and RxR).

Then for each point x in U, we can find an open disc Dx about x small enough so that it fits in U. If we like, we can also find a "rectangle" Rx small enough to fit inside Dx (top figure):

attachment.php?attachmentid=28528&stc=1&d=1285474877.png


Do you understand how the product topology in RxR naturally makes "open rectangles" open sets? It should follow very quickly from the definitions. What happens, now, if we take the union over all x in U of the rectangles Rx?

Similarly, for any rectangle, we can find an open disc inside it (bottom figure). Is this clear?
 

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1. What is a topological space?

A topological space is a mathematical concept that describes the arrangement of points or elements in a set, taking into account their proximity and connectivity. It is a foundational concept in topology, which is a branch of mathematics that studies the properties of space and spatial relationships.

2. What is a basis in a topological space?

A basis in a topological space is a set of open sets that can be used to define and generate the entire topology of that space. It serves as a building block for the open sets in a topological space and allows for the characterization of open sets in terms of basis elements.

3. How is a topological space different from a metric space?

While both topological spaces and metric spaces are mathematical constructs that describe spatial relationships, they differ in the types of properties they focus on. A metric space relies on a distance function, or metric, to define the notion of proximity between points, while a topological space focuses on the concept of open sets and their relationships.

4. What are some common examples of topological spaces?

Some common examples of topological spaces include the real line, Euclidean spaces, and the plane. Other examples include the spaces of continuous functions, topological groups, and manifolds. Topological spaces can also be constructed from more abstract sets, such as the set of all subsets of a given set.

5. How are basis elements related to open sets in a topological space?

In a topological space, every open set is a union of basis elements, and every basis element is an open set. This means that the basis elements form a basis for all open sets in the topological space. This relationship is important because it allows for a more concise way of describing and understanding the topology of a space.

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