Question about a collection of sets in the plane.

In summary, the collection of intervals of the form {\{a\}\times(b,c)\subset\mathbb{R^2}|a,b,c\in\mathbb{R}} is a basis for a topology on \mathbb{R^2}. This means that any open set in \mathbb{R^2} can be expressed as a union of these vertical intervals. This is achieved by letting a vary across the real line and pairing it up with all points on that line, creating open intervals going up and down from point a starting at b and going vertically up to c. This results in an open interval with first coordinate a and second coordinate between b and c.
  • #1
cragar
2,552
3

Homework Statement


Show that the collection
[itex] \{ \{a\}\times(b,c) \subset \mathbb{R^2} |a,b,c \in \mathbb{R} \} [/itex]
of vertical intervals in the plane is a basis for a topology on [itex] \mathbb{R^2} [/itex]

The Attempt at a Solution


My question is just really about (a)X(b,c)
am I just basically letting a varying across the real line and then just pairing it up
with all the points on that line. So at each vertical interval
it will be open interval going up and down from point a and starting at b and going up vertically to c.
 
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  • #2
It's the open interval whose points all have first coordinate a and the second coordinate lies between b and c. So yes, if that's what you mean.
 
  • #3
Hi. Could you explain your solution a bit more? I am stuck in the same problem..
 

FAQ: Question about a collection of sets in the plane.

What is a collection of sets in the plane?

A collection of sets in the plane is a group of sets that are contained within a two-dimensional space, such as a coordinate plane. Each set in the collection may contain multiple points or objects, and the sets may overlap or intersect in various ways.

What are some examples of collections of sets in the plane?

Examples of collections of sets in the plane include Venn diagrams, geometric shapes on a coordinate grid, and maps of countries or cities. These collections are often used to represent relationships between different sets or to visualize data.

How are collections of sets in the plane related to set theory?

Collections of sets in the plane are closely related to set theory, which is a branch of mathematics that deals with the properties and operations of sets. Set theory provides the foundation for understanding collections of sets in the plane and how they can be manipulated and analyzed.

Can collections of sets in the plane be used in real-world applications?

Yes, collections of sets in the plane have many real-world applications. For example, they can be used in computer science for data visualization, in statistics for data analysis, and in engineering for geometric modeling and simulations.

How are collections of sets in the plane different from collections of sets in other dimensions?

Collections of sets in the plane are different from collections of sets in other dimensions, such as 3D space, in that they are limited to two dimensions. This means that sets in the collection can only contain points with two coordinates, and the relationships between sets can only be visualized in two dimensions.

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