Is the Subset D={(x,y)| x≠0 and y≠0} an Open Set in R^2?

  • Thread starter Deadward1994
  • Start date
In summary, the conversation discusses how to prove that the subset D={(x,y)| x≠0 and y≠0} is an open set in R^2. The conversation also mentions the definition of an open set and suggests drawing and proving it analytically. The main focus is on finding a radius r that satisfies the conditions for all points in the open ball around (x,y) and the interesting case when (x,y) is close to (0,0). It is also mentioned that finding r for points close to (0,0) is more challenging and the question of what happens exactly at (0,0) is raised.
  • #1
Deadward1994
2
0

Homework Statement


Show that the subset D={(x,y)| x≠0 and y≠0} is an open set in R^2
.

Homework Equations


Open set: U is a subset of R^n. U is an open set when for every point X1, contained within U, there exists some open disk centered around X1 with radius r>0, that is completely contained within U. Or for simplicity's sake, a set U is open if it does not contain any of its boundary points.

The Attempt at a Solution


I have an understanding of what makes up an open set and know why this set is open, but i have no idea as to how I am meant to prove this, graphically or analytically( ideal method).
 
Last edited:
Physics news on Phys.org
  • #2
Deadward1994 said:

Homework Statement


Show that the subset D={f(x; y)| x≠0 and y≠0} is an open set in R^2
.

Homework Equations





The Attempt at a Solution


I have an understanding of what makes up an open set and know why this set is open, but i have no idea as to how I am meant to prove this, graphically or analytically( ideal method).
What does the notation f(x;y) mean?
 
  • #3
Sorry about that, I miss quoted the question
 
  • #4
First, tell us the definition of an open set.
 
  • #5
So let (x, y) be a point in D. Can you find a radius r such that all points in the open ball of radius r around this point are in D?

First you may want to try drawing this, then proving it analytically. Note that for most points it's pretty trivial - the interesting case is for when (x, y) is close to (0, 0).
 
  • #6
CompuChip said:
So let (x, y) be a point in D. Can you find a radius r such that all points in the open ball of radius r around this point are in D?

First you may want to try drawing this, then proving it analytically. Note that for most points it's pretty trivial - the interesting case is for when (x, y) is close to (0, 0).

Also, presuming you can always find ##r## for arbitrary ##(x,y) ≠ (0,0)##, what happens at ##(0,0)## exactly?

Get reeeeeally close to ##0##. Can you always find an ##r## such that ##(0,0)## is not contained any neighborhood of your point?
 
Last edited:

FAQ: Is the Subset D={(x,y)| x≠0 and y≠0} an Open Set in R^2?

What does it mean for a subset to be open?

In topology, a subset of a topological space is considered open if every point in the subset has a neighborhood contained entirely within the subset. This means that there are no boundary points or points on the edge of the subset.

How do you prove that a subset is open?

To prove that a subset is open, you need to show that for every point in the subset, there exists a neighborhood of that point that is entirely contained within the subset. This can be done by explicitly constructing the neighborhood or using a proof by contradiction.

What is the importance of proving that a subset is open?

Proving that a subset is open is important in topology because it allows us to identify properties of the topological space and its subsets. It also helps us understand the structure of the space and its relationship with other subsets.

Can a subset be both open and closed?

Yes, a subset can be both open and closed. This type of subset is known as a clopen set and it exists in certain topological spaces, such as discrete spaces or the empty set in any topological space.

What are some common techniques for proving that a subset is open?

Some common techniques for proving that a subset is open include using neighborhood definitions, proving by contradiction, using topological axioms, and using results from previous theorems. It is also helpful to have a good understanding of basic topological concepts and properties.

Similar threads

Proving Irregularity of {(x, y, z) within R^3 : x^2 + y^2 - z^2 = 0}
Replies
14
Views
2K
Show that if d is a metric, then d'=sqrt(d) is a metric
Replies
8
Views
3K
Proving intersection of finitely many open sets is open
Replies
1
Views
1K
Verification that ZxZ has no cluster points in RxR
Replies
3
Views
1K
Is the set open, closed, neither, bounded, connected?
Replies
7
Views
2K
Exploring Open, Closed, Bounded, and Compact Sets in R with a Unique Metric
Replies
1
Views
1K
Difference between closed set and open set is a closed set
Replies
9
Views
5K
Prove Set U is Open | Product Topology on R^2
Replies
17
Views
2K
Continuity of a function under Euclidean topology
Replies
2
Views
2K
Is Set S Open in R3? A Proof by Using Open Discs
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top