Connectedness of coordinates with one rational point

In summary, the conversation discusses proving that the collection of all points in R^2 with at least one rational coordinate is connected. The speaker suggests using paths consisting of horizontal and vertical segments to prove that it is path-connected. They also mention finding a way to generalize this idea.
  • #1
hypermonkey2
102
0
Hi all, i found this problem in a topology book, but it seems to be of an analysis flavour. I'm stumped.

Show that the collection of all points in R^2 such that at least one of the coordinated is rational is connected.

My gut says that it should be path-connected too (thus connected), but I am finding the proof elusive... any thoughts?

cheers
 
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  • #2
It is path-connected. Try paths consisting of horizontal and vertical segments moving along straight lines. For instance, to move from (0, √2) to (π, 1/2), you could first move along the straight line segment from (0, √2) to (0, 1/2), and then along the straight line segment from (0, 1/2) to (π, 1/2). Now find a way to generalize that line of thought.
 

1. What does the term "connectedness of coordinates" mean?

The connectedness of coordinates refers to the relationship between different sets of coordinates on a graph or in a mathematical equation. It is a measure of how closely related or dependent the coordinates are on each other.

2. How is the connectedness of coordinates related to rational points?

Rational points are coordinates that can be expressed as a ratio of two integers. The connectedness of coordinates with one rational point means that there is a rational point that is shared by all the other coordinates, showing a strong connection or dependence between them.

3. Why is the connectedness of coordinates with one rational point important in mathematics?

This concept is important because it helps us understand the relationship between different sets of coordinates and how they are connected. It also allows us to make predictions and solve problems by using the known rational point to find other unknown coordinates.

4. How can the connectedness of coordinates with one rational point be determined?

The connectedness of coordinates can be determined by analyzing the patterns and relationships between the coordinates, looking for common factors or similarities. It can also be determined by using mathematical tools and formulas to find the most rational point that connects all the coordinates.

5. What are some real-world applications of the connectedness of coordinates with one rational point?

This concept has many practical applications, such as in navigation systems, where coordinates are used to determine the location of a point on a map. It is also used in engineering and construction projects to accurately plot and connect different points. Additionally, it is used in financial analysis to identify relationships between different data points.

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