Intersection of sets in ##\Bbb{R}^2## and Open Maps

In summary, the homework statement is that the canonical projection onto the first coordinate of ##\pi_1## is not an open map. However, this map is still defined on the subspace of closed intervals that are half-open and open.
  • #1
Bashyboy
1,421
5

Homework Statement



No problem statement

Homework Equations

The Attempt at a Solution



Let ##A = \{(x,y) ~|~ x \ge 0 \mbox{ or } y=0 \}## be a subspace, which can be shown closed, of ##\Bbb{R}^2##. If my calculations are right, isn't ##A \cap [(0,\infty) \times \Bbb{R} ] = (\{0\} \times \Bbb{R}) \cup ((0,\infty) \times \{0\})## and therefore ##\pi_1(A \cap [(0,\infty) \times \Bbb{R} ] ) = [0,\infty)##, where ##\pi_1 : \Bbb{R}^2 \to \Bbb{R}## the canonical projection map onto the first coordinate, thereby showing that ##\pi_1## restricted to ##A## is not an open map?
 
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  • #2
##(0,0)## isn't an element of your second set ## (0,\infty) \times \Bbb{R} ##.
Bashyboy said:
##A \cap [(0,\infty) \times \Bbb{R} ] = (\{0\} \times \Bbb{R}) \cup ((0,\infty) \times \{0\}) ##
 
Last edited:
  • #3
Bashyboy said:

Homework Statement



No problem statement

Homework Equations

The Attempt at a Solution



Let ##A = \{(x,y) ~|~ x \ge 0 \mbox{ or } y=0 \}## be a subspace,
But it isn't a subspace.
 
  • #4
WWGD said:
Why not, isn't ##0 ## in ## \mathbb R ##?
Yes, but why is it in ##(0,\infty)##?
 
  • #5
fresh_42 said:
Yes, but why is it in ##(0,\infty)##?
Never mind, let me delete. I was somehow thinking of intervals. The notation ##(a,b)## used both for points and intervals can be confusing.
 
  • #6
WWGD said:
Never mind, let me delete. I was somehow thinking of intervals. The notation ##(a,b)## used both for points and intervals can be confusing.
Yes, therefore I like ##]0,\infty[## for open intervals a lot more. But I seem to be the only one here.
 
  • #7
fresh_42 said:
Yes, therefore I like ##]0,\infty[## for open intervals a lot more. But I seem to be the only one here.
What do you use for closed intervals?
 
  • #8
##[a,b]## closed, ##[a,b[## half-open and ##]a,b[## open. I wonder where the round ones came from.
 
  • #9
Ah! You are right. I think I know how to solve the problem now, which is to find a set that is open in ##A## such that its image under ##q## is not open in ##\Bbb{R}##. The idea is to intersect the ##y##-axis with an open ball. More specifically, consider ##A \cap B((0,2),1) = \{(x,y) \in A \mid x^2 + (y-2)^2 < 1 \}##. I claim that the image of this is ##[0,1)##. If ##(x,y) \in A \cap B((0,2),1)##, then ##x \ge 0## or ##y=0## and ##x^2 + (y-2)^2 < 1##, which implies ##|x| < 1##. If ##x \ge 0##, then ##q(x,y) = x \in [0,1)## (note: the ##y=0## case cannot obtain). Now, if ##x \in [0,1)##, then ##q(x,2) = x## where #(x,2) \in A \cap B((0,2),1)##.

How does this sound?
 

1. What is the intersection of sets in ##\Bbb{R}^2##?

The intersection of sets in ##\Bbb{R}^2## refers to the common elements shared by two or more sets in a two-dimensional coordinate system. It is the point or points where the sets overlap.

2. How do you find the intersection of sets in ##\Bbb{R}^2##?

To find the intersection of sets in ##\Bbb{R}^2##, you can graph the sets on a coordinate plane and see where they overlap. Another way is to solve the equations or inequalities that define each set and find the values that satisfy both.

3. What is an open map?

An open map is a function between two topological spaces that preserves the concept of openness. In other words, an open map takes open sets to open sets.

4. How do open maps relate to the intersection of sets in ##\Bbb{R}^2##?

In the context of ##\Bbb{R}^2##, open maps can help determine the intersection of sets by preserving the openness of sets. If the sets are open, the intersection will also be open, making it easier to find and work with.

5. What are some applications of the intersection of sets in ##\Bbb{R}^2## and open maps?

The intersection of sets in ##\Bbb{R}^2## and open maps have various applications in mathematics, physics, engineering, and computer science. They are used in optimization problems, topology, graph theory, and more. In real-world scenarios, they can help determine the common regions of overlapping data or areas of interest in a map.

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